Department of Chemical Engineering, June 20, 1995

The Deformation of Cellular Entities

by

Kuo-Kang Liu

Thesis submitted for the degree of

Doctor of Philosophy

University of London

Department of Chemical Engineering,

Imperial College of Science,Technology and Medicine

London

June 20, 1995

(

"If anyone thinks that he knows anything, he has not yet

come to know as he ought to know."

1 Corinthians (The Bible): Chapter 8; Verse 2

Preface

This dissertation is a description of the work carried out in the Department

of Chemical Engineering and Chemical Technology, Imperial College, London

between June 1993 and June 1995. Except where acknowleged, the material

presented is the original work of the author and includes nothing which is the

outcome of work done in collaboration, and no part of it has been submitted for

a degree at any other University.

I am deeply grateful to Professor B. J. Briscoe for his excellent supervision

during the course of my research. His help and guidance have been invaluable.

The author is also greatly indebted to both Professor D. Tabor, F.R.S., and Pro-

fessor K. L. Johnson, F.R.S., for their many invaluable suggestions concerning

the deformation of elastomeric spheres. It has also been a pleasure to receive help

from Dr. D. R. Williams in the setup and maintenance of the equipment which

was originally designed by him. I would also like to thank Dr. P. F. Luckham for

his kind assistance during discussions. The experimental assistance from Mr. G.

Corfield for the preparation of elastomeric spheres is acknowledged. The provi-

sion of microcapsules by Mr. D. J. Brown (Zeneca) of tomato cells by Professor

P. Lillford and Dr. T. Foster (Unilever) is also gratefully acknowledged.

I would also like to acknowledge the DTI Link Colloid Research Programme

for supporting this project. I would also like to especially thank Taiwanese

government for the provision of a Life Science Scholarship.

Kuo-Kang Liu

June, 1995

1

Abstract

This thesis describes both an experimental and a theoretical investigation into

the interfacial energy and intrinsic mechanical properties of various single micro-

cellular entities (ca 65-.500 im diameter) , through the investigation of their

compressive deformability. A novel experimental technique, which involves the

upsetting (compression of single micro-cellular entities between two parallel

plates), has been developed to directly measure the imposed forces and the corre-

sponding compressive displacements simultaneously. This apparatus has a force

and displacement resolution of 10 jN and 0.25 jim, respectively. The apparatus

also allows the optical measurement of the major features of the deformation

geometry, such as the central lateral extension, which is necessary to properly

identify the deformation behaviour of these systems. The theoretical studies un-

dertaken include, both analytical and numerical analyses, which are developed

for interpreting the experimental data. Specifically, theoretical analyses will be

presented for both linear and non-linear elastic compressive deformation of ho-

mogeneous solid spheres, as well as liquid-filled spherical membrane systems, at

both small and large imposed strains.

In general, the experimental data and the theoretical predictions are shown

to be in good accord for both the homogenous elastomeric spheres and the liquid-

filled microcapsules with an elastomeric membrane. However, for the biological

cells the constitutive equations (strain-stress relationships) for the cell walls are

2

3

not satisfactorily resolved. The autoadhesive force is shown to be a significant

factor for the description of the deformation of the homogeneous elastomeric

micro-spheres at small deformations. An implementation of an established the-

oretical analysis (JKR theory) for small deformations provides an estimate of

the interfacial free energy between the single elastomeric micro-sphere and the

compressive platens. The application of large deformation theories, either the

Tatara model for the homogeneous elastomeric spheres or the membrane model

for the microcapsules, allows the elasticity of a single particle to be accurately

determined. The Tatara based analysis, combined with experimental data at

40 % deformation, allows the explicit estimation of the Poisson ratio for the

elastomeric spheres.

Contents

List of Figures 8

List of Tables 15

Nomenclature............................... 16

1 General Introduction 21

1.1 Introduction .............................21

1.2 The aim of this study ........................24

1.2.1 The Methodology ......................26

1.3 The structure of the Thesis ....................29

2 Experimental methods 30

2.1 Review of the previous methods ..................30

2.1.1 Nano-/Micro-indentation ..................30

2.1.2 Micropipette aspiration method ..............32

2.1.3 Compression method ....................34

2.2 Micro-upsetting method ......................36

2.2.1 Equipmental setup .....................36

2.2.2 The major merits of the current method .........44

2.3 Instron Universal Testing Machine .................46

I

CONTENTS

5

3 Theoretical models 47

3.1 Review of the theories .......................47

3.2 The identification of applicable theories ..............50

4 The deformation of homogeneous micro-elastomeric spheres 54

4.1 Introduction ............................. 55

4.2 The theories of deformation .................... 58

4.2.1 The theories without autoadhesion effects

58

4.2.1.1 Hertz theory ................... 58

4.2.1.2 Tatara theory .................. 59

4.2.2 The theories with autoadhesion effects .......... 64

4.3 Experimental ............................ 70

4.3.1 Material preparation .................... 70

4.3.2 Experimental procedure .................. 70

4.4 Results and discussions ....................... 72

4.4.1 Autoadhesion effects .................... 72

4.4.2 Mechanical responses .................... 80

4.4.2.1 Compressive compliance ............. 80

4.4.2.2 Scaling effects ................... 84

4.4.2.3 Geometric features of deformations ....... 84

4.5 Summary ..............................91

5 The deformation of liquid-filled micro-cellular entities 93

5.1 Introduction ............................. 94

5.2 Experimental ............................ 96

5.2.1 Material ........................... 96

5.2.2 Experimental procedure .................. 96

5.3 The theoretical analysis ...................... 98

CONTENTS

6

5.4 Results and Discussions ......................104

5.4.1 Elastic modulus of the membrane .............104

5.4.2 Internal pressure ......................109

5.4.3 Membrane tension .....................112

5.4.4 Geometric features of the deformation ..........114

5.5 Summary ..............................119

6 The deformation of biological cells 121

6.1 Introduction .............................121

6.2 Experimental ............................124

6.2.1 Material ...........................124

6.2.2 Stress-relaxation experiment ................126

6.3 Theoretical analysis .........................126

6.3.1 Constitutive equations ...................126

6.3.1.1 General constitutive equations for cell membrane 126

6.3.1.2 Specific constitutive equations for plant cell 128

6.3.2 Governing equations for compressive compliance .....129

6.3.3 Stress Relaxation analyses .................130

6.4 Results and discussions .......................132

6.4.1 Stress relaxation ......................132

6.4.2 Compressive compliance and bursting phenomena . . 134

7 Conclusions 139

7.1 The governing effects responsible for the observed deformation 140

7.2 The experimental method .....................142

7.3 The theoretical interpretation ...................143

7.3.1 For the homogeneous elastomeric spheres .........143

7.3.2 For the liquid-filled membrane ...............143

CONTENTS 7

Appendix 145

A The jump phenomenon 145

A.1 Long range forces: Lifshitz theory .................145

A.2 The experimental configuration and the origin of the "jump" 146

A.3 The Johnson method ........................147

B Computer programs 150

References 162

List of Figures

1.1 A flow chart outlining the methodology of the current work and

thepotential applications ......................27

2.1 The two methods of indentation hardness measurement: (a) the

imaging method, when the diameter or diagonal, d, is measured

after indentation; and (b) the compliance method, where the re-

action force, F, and the depth of penetration, h, are measured

continuously during the indentation. Adopted from Sebastian

(1994), with permission.......................33

2.2 Micropipette aspiration method: is the pressure difference

between the pipette interior and the outside medium, and L is

the length of the aspirated projection of the membrane......35

2.3 The compression method: F is the force on each plate where the

plates are separated by each other a distance Z..........37

2.4 The schematic view of the micro-upsetting instrument (not to

scale).................................39

2.5 The schematic view of the computer-controlled systems of the

micro-upsetting instrument (not to scale). The items outside the

boxed region marked as "Host computer" were mounted upon a

vibration isolation table.......................40

8

LIST OF FIGURES

9

2.6 The deflection of the cantilever beam under large imposed loads

(not to scale). zX is the maximum deflection distance and 0 is

the rotation angle of micro-platen caused by the deflection. . . 42

2.7 (a) A photograph of the arrangement of two parallel platens: the

indenter (micro-platen) and the petri dish (see the following dia-

gram for details). (b) An idealised schematic view of the platens

of the micro-upsetting instrument (not to scale). LY is the maxi-

mum error in the measured displacement caused by the imperfect

parallelism of these two platens, and dy is the real error arising

from this source. L is the diameter of the indenter and D is the

diameter of the contact area between the deformed cellular entity

andthe platens............................ 43

3.1 The general features of the loading! unloading curves for three

different types of structures: (a) homogeneous elastomeric sphere;

fracture stress is assumed to be extremely difficult to achieve; that

is the sphere does not fracture under the maximum loading force.

(b) liquid-filled spherical elastomeric membrane; (c) liquid-filled

spherical elastomeric shell when wall rupture occurs.......51

3.2 The various theories available for the descriptions of the sequen-

tial stages, appearing in the loading and unloading, for the elas-

tomericmicro-spheres ........................53

LIST OF FIGURES

10

4.1 Schematic of the symmetrical shape of a soft elastic sphere corn-

pressed, at large stains, between two flat rigid surfaces, and the

movement of a point C(z, r) on the surface before deformation to

C'(z',r') after deformation, where U(z,r) and U(R) indicate the

lateral extensions at the contact surface and the central diameter,

respectively. The c is the approach; a' is the computed radius

of the contact surface at the vertical position (z); a is the radius

before deformation. (adapted from Tatara's original paper (1991)) 63

4.2 The flow chart for the enhanced version of the algorithm for solv-

ing the set of the equations of the Tatara theory used in the

currentstudy.............................65

4.3 (a) Deformable (elastic) spheres on a rigid surface in the absence

(Hertz) and presence (JKR) of adhesion. (b) elastic adhering

sphere about to separate spontaneously from an adhesive contact.

Adapted from Israelachvili (1991), by permission.........68

4.4 Typical loading/unloading curve for the small deformation ex-

periment. Starting (S); contacting (A); loading (AB); unloading

(BC); pull off (CD); ending (E). The members "jump" (see Ap-

pendix A) into contact at A. At A' the applied load is zero. The

pull-off force corresponds to the difference between the magnitude

of the forces at points C and D...................74

4.5 The comparison of the theoretical (predicted by Hertz and JKR

theories) and experimental unloading curve for small dimension-

less approaches (c*). Case A; translated experimental data (by

shifting a displacement of 0.6 pm) for the pull-off speed 1 pm

sec 1 , Case B; experimental data for the pull-off speed 5 pm sec 1 . 79

LIST OF FIGURES

11

4.6 The comparison of experimental and theoretical load versus di-

mensionless approach, a, curves for a 270 m poly(urethane)

sphere. The experimental data and various predictions are shown.

The Young's moduli are respectively 2.25 MPa for the Hertz pre-

dictions and 2.06 MPa for the Tatara analysis. The Poisson's

ratiois taken as 0.48.........................82

4.7 Comparison of the theoretical and the experimental dimensionless

applied load, P/K, versus dimensionless approach (cf) curves for

300 im and 38.3 mm poly(urethane) spheres...........85

4.8 Comparison of the theoretical and the experimental dimensionless

central lateral extension, U*(R) , versus dimensionless approach,

curves for 240 im and 38.3 mm poly(urethane) spheres. . . 87

4.9 Photographs of the deformed elastomeric sphere (ca 240 tim) from

a bottom view for various dimensionless approaches (a); (a) a=0

% (b) a 9.5 % (c) a= 28.3 % (d) a= 55.8%...........88

4.10 Simulated dimensionless central lateral extension, U*(R), as a

function of dimensionless approach, a, for various Poisson ratios

(ii) computed by the modified Tatara analysis...........90

5.1 Schematic representation of the microcapsules...........97

5.2 Geometry for the contact problem for the half of a thin wall spher-

ical membrane between two large rigid plates...........103

5.3 The flow chart for the enhanced version of the algorithm for solv-

ing the set of the equations of the membrane model used in the

currentstudy.............................105

LIST OF FIGURES

12

5.4 The experimental loading and unloading (after bursting) curves

of a 65 ,am microcapsule. The bursting point is near 58 % dimen-

sionless approach, (1 - i/ro))................... 107

5.5 Photographs of the bottom view of a deformed microcapsule for

various dimensionless approaches (y); (a) y=O % (b) y= 20 % (c)

y= 40 % (d) y= 58 %........................108

5.6 The dimensionless experimental loading/unloading curve (defor-

mation up to 40 %) and theoretical predictions produced by the

membrane model with a Mooney- Rivlin material law. The param-

eter y = (F/Ci hr0)) is the dimensionless force and the quantity

(1 - /r0 ) 3 ) is the dimensionless approach. is 1.0 and C1 is

16.08 MPa..............................110

5.7 The dimensionless experimental loading/unloading curve (defor-

mation up to 40 %) and theoretical predictions produced by the

membrane model with Neo-Hookean material law. The parame-

ter y = ( F/Ci hr0)) is the dimensionless force and the quantity

(1 - i1/r0 X,) is the dimensionless approach. is 1.0 and C1 is

16.14 MPa..............................111

5.8 Internal pressure versus deformation curve for the compression of

a compressive microcapsule. The Young's moduli of the mem-

brane are respectively 2.69 MPa for a Neo-Hookean material and

2.68 MPa for a Mooney-Rivlin response..............113

LIST OF FIGURES

13

5.9 Variation of the wall tension, the extensional force per unit wall

thickness, with angular position, b, for a 58 % deformation. The

Young's moduli of membrane are respectively 2.69 MPa for a Neo-

Hookean material and 2.68 MPa for a Mooney-Rivlin system. T1

and T2 are the stress resultants in the meridional and circumfer-

ential directions, respectively....................115

5.10 The simulated deformed shapes of a microcapsule () = 1, fi =

0.1) for 20 %, 40 % and 60 % deformation as used in the current

study.................................116

5.11 A comparison of the computed geometric features of the deforma-

tion of a homogeneous elastomeric sphere with a Poisson ratio of

0.5 and a liquid-filled spherical entity with an elastomeric mem-

brane; (a) the dimensionless central lateral extension versus the

dimensionless approach (b) the dimensionless contact radius ver-

sus the dimensionless approach ..................118

6.1 Schematic representations of typical structures of cells: (a) plant

cell(b) animal cell..........................123

6.2 Photographs of the single isolated tomato cells: (a) green cell (ca

250 iim diameter) (b) red cell (ca 250 m diameter).......125

6.3 Three mechanical models for a viscoelastic material. (a) a Maxwell

body, (b) a Voigt body, and (c) a Kelvin body (a standard linear

solid). Adopted from Fung (1993) with permission........131

6.4 Relaxation behaviours of (a) a Maxwell, (b) a Voigt, and (c) a

standardlinear solid.........................133

6.5 The experimental results and theoretical correlation of the stress

relaxation for two types of tomato cells (a) green cell (b) red cell. 135

LIST OF FIGURES

14

6.6 Typical experimental curves of the compressive compliance for

two types of tomato cells; (a) green cell (b) red cell........136

6.7 The theoretical predictions for the compressive compliance for the

spherical cells, obtained by using the membrane model combined

with the selected constitutive equations for the cell membrane. . 137

A.1 (a) The compressive load (P)-deflection (u) produced by the JKR

theory for the interplation of the jump phenomenon (Johnson

1995; private communication). (b) The actions of the sphere and

platen at (i) the point Q and (ii) the point D ..........148

List of Tables

1.1 The characteristics of the investigated micro-cellular entities . . 25

3.1 Various models available for the prediction of the deformations of

micro-cellular entities under compression and indentation . . . . 49

4.1 Effects of strain (dimensionless approach) and strain-rate (pull-

off speed) on the adhesion of microscopic poly(urethane) spheres

(300 jim) in contact with two glass plates .............75

5.1 The comparison between the experimental and theoretical dimen-

sionless central lateral extension versus dimensionless approach

relationships.............................117

15

Nomenclature

Chapter 2

D Diameter of contact area between deformed cellular entity and platen

d Diameter or diagonal of the indentation image

F Applied force (compression method)

h Depth of penetration (indentation method)

L Diameter of the indenter (micro-platen)

L Length of the aspirated projection of the membrane

P Reaction force of identation (indentation method)

by the cantilever deflection

Z Distance between two parallel platens (compression method)

greek symbols:

AP Applied pressure difference (micropipette method)

LX Maximum deflection distance of cantilever beam

Maximum error in the measured displacement

0 Rotation angle of micro-platen

Chapter 4

A Hamaker non-retarded force constant

a Radius of contact area

a' Computed radius of the contact surface by Tatara theory

16

Nornenclat ure 17

E

E0

H0

K'

P

Pa

PC

P1

P

R

u(z, r)

U(z,a)

U(R)

U*(R)

w(z, r)

z

Young's modulus

Initial Young's modulus (Tatara theory)

"Jump" distance

[_4ER2- [3(1_2)

Contact normal load

Effective force of adhesion

Pulling force or the force of attraction

Apparent Hertz load by including the auto adhesive force

P3Ririy

Radius of the sphere

Radial displacement of half-space elastic body

Lateral extension predicted by Tatara theory

Lateral extension at the central diameter

Dimensionless central lateral extension ( U(R)/R)

Vertical compressive displacement of half-space elastic body

Vertical position

greek symbols:

a Approach

Dimensionless approach (the ratio of a to R)

Asperity radius

Strain

F

Measured work of adhesion

71 Surface free energy of body 1

72 Surface free energy of body 2

712 Interfacial energy of interface 12

w

Thermodynamic work of adhesion

a*

c1&c2

D

E

F

h

h

I

P

r

T

U(R)

Nomenclature

0 "Adhesion prameter" (proposed by Tabor)

Dimensionless parameter for distinguishing the regimes

of applicability of DMT & JKR theories

11

Poisson's ratio

0• Distribution of the asperity bight

o(x)

The stress distribution within the contact area

Subscripts

jkr Variables predicted by JKR theory

t Variables predicted by Tatara theory

z Variables predicted by Hertz theory

H Higher values of variables in the two points & circulation method

M Median values of variables in the two points & circulation method

L Lower values of variables in the two points & circulation method

Chapter 5

Dimensionless radius of contact area

Two material contants (for the strain energy function)

Bending rigidity of shell

Young's modulus

Reaction force of deformed spherical membrane

Initial wall thickness of shell

Thickness of plate and shell

Strain invariant (for the strain energy function)

Internal pressure of microcapsules


Stress resultant

Dimensionless central lateral extension ( U(R)/R)

Nomenclature 19

W =6'

W* Strain energy function for Mooney-Rivlin law

greek symbols:

Ratio of C2/C1

S

=A2sinW

11 Distance between rigid plate and equator of spherical membrane

A principal stretch ratio

F

Angle of contact area (membrane model)

"

Possion's ratio

0. Principal stress

Central expanded radius of spherical membrane

4,

Angular position (membrane model)

Superscripts

Variables differentiated with respect to J1

Subscripts

o Values of variables before deformation

Initial values of variables

1 Variables in the meridional direction

2 Variables in the circumferential directions

Chapter 6

B&C

Two material contants (for the strain energy function)

D

Bending rigidity of shell

E

Young's modulus

F

Reaction force of deformed spherical membrane

Nom en ci at ure 20

h

Initial wall thickness of shell

I

Strain invariant (for the strain energy function)

P

Internal pressure of microcapsules

r


T

Stress resultant

wc*

Strain energy function developed by Chaplain.

w;

Strain energy function developed by Skalak et al.

greek symbols:

S

=\2sin4'

17 Distance between rigid plate and equator of spherical membrane

principal stretch ratio

F

Angle of contact area (membrane model)

71

Possion's ratio

/7 Shear modulus

0• Principal stress

T

Ratio of B/C

Central expanded radius of spherical membrane

4'

Angular position (membrane model)

Superscripts

Variables differentiated with respect to 4'

Subscripts

o Values of vraiable before deformation

s Initial values of variables

1 Variables in the meridional direction

2 Variables in the circumferential direction

Chapter 1

General Introduction

The main purpose of this Chapter is directed to defining the problem which

has been investigated and thus explaining the objectives of the current work.

A preliminary survey of the potential applications which may be derived from

this study is also recorded. In addition, the main theme of this Thesis, a novel

methodology, which includes both theoretical and experimental approaches is

described briefly. The Chapter finally concludes by giving a review of the con-

tents of each chapter subsequently presented in this Thesis.

1.1 Introduction

It is now recognised that the deformation behaviour of single micro-cellular en-

tities is potentially important not only for characterising the bulk mechanical

responses of the aggregation of these cellular entities, such as particulate ag-

glomerates and biological tissues, but also for improving our understanding of

the rheology of concentrated dispersions. For example, the rheological properties

of concentrated cellular suspensions have now been recognised to be directly re-

lated to the deformability of the single cellular entities (Evans & Lips 1990) and

the interfacial interactions of these entities and their external surface (Briscoe

1994). Hence, data for the deformation of single micro-cellular entities can form

21

1.1 Introduction 22

the bases for discrete particle-particles interaction laws used in the construction

of rheological models. In addition, characterising a single biological cell provides

an essential insight into the mechanical deformation response of biological tis-

sues. However, various studies (Bliem 1989; Zhang et al. 1992) have pointed

out that the systematic experimental investigation of the deformability of sin-

gle micro-cellular biological entities and the interpretation of these experimental

data have both proved to be extremely difficult to achieve.

The origins of the forces between micro-cellular entities and surfaces, which

are well described in the literature, may be distinguished between the surface

forces and the bulk mechanical forces. The former are mainly contributed by

the van der Waals forces as well as electrostatic forces, and vary significantly

with the surface topography, the bulk viscoela.stic properties as well as the con-

ductivity of the materials. The latter forces are mainly governed by the bulk

mechanical properties such as the elasticity and Poisson ratios of the materials.

At small deformations, the autoadhesive forces may play a significant role in

the deformation of micro-cellular entities (especially for the homogeneous elas-

tomeric sphere in a dry contact), whilst at large deformations the mechanical

forces will be prevailing. The autoadhesive forces have been shown to be an

important effect for low loads such as those which occur in particle-particle

interactions in colloidal suspensions (Israelachvili 1991; Johnson 1993). How-

ever, the mechanical forces have been shown to be important in many aspects

of the processing and handling of particulate systems, such as the processes of

compaction and extrusion.

Because biological cells, due to their complex structures and their material prop-

erties, which are intrinsically difficult to describe, it is rational and sensible to

study the deformation behaviours of the well-defined artificial systems, such as

the homogeneous elastomeric spheres and the liquid-filled microcapsules, as a

1.1 Introduction 23

first step. Furthermore, these two model cellular systems, in view of the avail-

able theoretical analyses, may both be considered as the limiting cases of shell

models. These are described in the classic structural mechanics literature, which

considers that the deformations are mainly governed by both the bending forces

and stretching forces in the shell. The homogeneous elastomeric sphere may be

described as a shell which has the ratio of wall to radius of 1.0 ( thick

shell); whilst the microcapsule may be considered as a thin-shell (or membrane)

system in which the bending moment contributions may be neglected. Some

studies (Petersen et al. 1981; Evans & Skalak 1979) have pointed out that all of

these simplified models fail to accurately predicate the deformation behaviour

of biological cells. Currently, there is no suitable model available that allows

the accurate description of the deformation behaviour of biological cells in de-

tail. Both the membrane model and the homogeneous elastic sphere models

are widely used for describing biological cells in many studies (Hochmuth 1987;

Zahalak et al. 1990; Zhang et al. 1994). It is intuitively obvious that for the

compressive deformation of these three types of structures the applied loading

and unloading forces will vary in different ways. The general features of these

deformations are discussed in detail, from an universal point of view, in the

Chapter 3.

Since most micro-cellular entities, such as elastomeric particles, microcap-

sules as well as biological cells, have intrinsically low moduli (values of 0.1 MPa

to 10 MPa are common), large deformations are easy to achieve under small

external loads. Furthermore, for these materials the interrelationships between

strain and stress (constitutive equations) are usually non-linear. Hence, there is

a great need for appropriate theoretical models which include the large deforma-

tion formulation, with a non-linear elasticity. Taking into account these factors,

some advanced models, either for the homogeneous elastomeric spheres or for

1.2 The aim of this study 24

the liquid-filled membrane systems, are presented in the current study. These

higher-order models are more accurate than most of the previously reported

studies which have only applied first-order analyses.

It is obvious that the deformability of single micro-cellular entities is normally

governed by the interfacial, the mechanical and the viscoelastic properties, of

the entity. Hence, all of these properties may, in principle, be deduced from

the characterisation of the deformability of single micro-cellular entities. In the

current study the experimental method combined with appropriate theoretical

analyses, both analytical and numerical, has facilitated the determination of

some of these material properties which are shown in Table 1.1

1.2 The aim of this study

The main theme to be described in this Thesis is the development of a rigorous

methodology which combines both experimental and theoretical approaches in

order to facilitate the measurement of interfacial energy and intrinsic mechanical

properties of soft micro-particles (ca 65 500 gm), through the investigation

of their deformabilities (see Figure 1.1). To achieve this goal, in the experi-

mental aspect, a novel technique which involves the upsetting (compression of

single micro-cellular entities between two parallel plates) has been developed

and constructed to directly measure the imposed forces and the corresponding

compressive displacement simultaneously. An incorporated visualisation system,

which allows for the investigation of the corresponding major geometric defor-

mational parameters such as lateral extension as well as the failure phenomena,

has been developed and used in the current study. With respect to theoretical

aspects, both analytical and numerical analyses have been used to quantitatively

interpret the interrelationships between the force and the displacement as well as


Table 1.1 The characteristics of the investigated micro-cellular entities

Type of particlesSystem description(a)structure(b) compressibility(c)elasticityMaterial properties(a)mechanical

(b)viscoelastic(c) interfacial

elastomeric spheresCa. 300 pmhomogeneous solidincompressible*nonlinear

elasticity**Poisson ratio**

interfacial free energy

microcapsulesCa. 65 pmliquid-filled membraneincompressiblenonlinear

elasticity**

Poisson ratio**internal pressure**bursting strength**

plant cellsca. 500 pmcomplexunknownnonlinear

elasticityPoisson ratiointernal pressurebursting strengthstress relaxation**

* Nearly incompressible; Poisson ratio is approximately 0.497.

** The properties have been estimated in the current work.


the major geometric deformational parameters. This type of analysis provides,

in principle, a method for obtaining the critical physical information regarding

the mechanical and interfacial characteristics of the materials.

1.2.1 The Methodology

A rigorous methodology has been developed and is described by the author based

upon the conventional approaches to material characterisation (Fung 1993).

This methodology is applied to characterise the deformation behaviour of cel-

lular entities composed of various materials and structures, in order to explore

both their mechanical! viscoelastic and interfacial properties. Specifically, the

Thesis addresses the follow parts and some topics in a more detail than others.

1. The study of both the original and the deformed geometric configurations of

the particles.

2. The determination of the mechanical properties of single soft particles that

are involved in the problem. In particle characterisation, this step is often

very difficult, either because the size of investigated micro-cellular entities

is too small to be handled, or because the appropriate mathematical model

which includes reasonable boundary conditions and the formulation of in-

trinsic material properties is extremely difficult to develop. Furthermore,

as mentioned in the previous section, soft particles are often subjected to

large deformation, and stress-strain relationships (constitutive equations)

are usually nonlinearly and also history dependent. The nonlinearity and

time-dependence makes the determination of constitutive equations a chal-

lenging task. Usually however, one can determine the mathematical form

of the constitutive equations of the material quite readily, with certain

numerical parameters left to be determined by the experiments to be sub-


Single micro-cellular entitysystems

.

Experimentalinvestigation

The MethodologyOf Current Work

Theoreticalmodelling

Load-displacemen

Deformedcurve shape

Constitutiveequations

Characterising the

Computermechanical and simulation

interfacial properties

Multiple micro- \cellular entitysystemsS.

Correlating the deformationof a single cell to the bulkdeformation of tissue

Potential Applications

Exploring the particle-particle interaction lawsused in the construction ofrheological models

Figure 1.1 A flow chart outlining thc methodology of the Current workand the potential applications


sequently mentioned in items (6) and (7); see below.

3. In contact adhesion for small homogeneous elastomeric spheres, at small

deformations, the interfacial properties may play an important role on the

response of the particles to deformation. A well-known autoadhesive the-

ory, Johnson-Kendall-Roberts (JKR) theory (Johnson et al. 1971) , has

been applied to the current study. Based upon the experimental data for

the "pull-off" and the "jump contact" experiments, some important inter-

facial properties may be determined. In turn, the possible autoadhesive

mechanisms, such as van der Waals and electrostatic forces, have been

examined.

4. An examinationjhe external environment in which the possible mechanisms

may be involved, in order to obtain meaningful boundary conditions; for

example, the roughness on the contact surface between the particle and

plates.

5. The solution of the mathematical formulation (governing equations with ap-

propriate boundary conditions) by analytical method or numerical,

6. The implementation of experiments that test the solutions of the problems

mentioned above. Then as necessary, reformulation and resolution of the

mathematical problem to ensure that the results of theory and experi-

ment do correspond to each other, i. e., that they are testing the same

hypotheses.

7. A comparison of the experimental results with the corresponding theoretical

ones. By means of the comparison, the determination that the hypotheses

made in the theory are justified, and, if they are, the numerical values of

1.3 The structure of the Thesis 29

the undetermined coefficients in the constitutive equations are determined.

Hence, some of the material properties of the investigated entities may be

estimated.

8. And finally and in general, once a theory is so validated, it may be applied

to predict the outcome of other problems associated with the same basic

equations. Then one may use the method to explore practical applications

of the theory and experiments.

1.3 The structure of the Thesis

The rest of the Thesis consists of six chapters. In Chapter 2, both the previous

experimental techniques and the current method, are reported in detail. Chapter

3 briefly describes the various theories which are adopted in the current study.

Both the experimental data , associated with appropriate theoretical analysis,

on the deformation behaviours of the homogeneous elastomeric spheres are pre-

sented in Chapter 4. Apart from the mechanical response, the autoadhesive

force, which is shown to be a significant factor for the elastomeric spheres at

small deformations, is also discussed in this Chapter. The results for the defor-

mation of artificial micro-cellular entities, the liquid-filled microcapsules with an

elastomeric membrane, are presented in Chapter 5. In this Chapter the charac-

terisation of the elasticity and the bursting strength of the membrane, as well as

the tension forces on the membrane, are reported. Chapter 6 describes some pre-

liminary results on the deformation of biological cells; two types of tomato cells.

The viscoelastic behaviours of these tomato cells are also addressed. Finally,

some of the more important conclusions are given in Chapter 7.

Chapter 2

Experimental methods

This Chapter is composed of two major parts: The first part discusses some

of previous experimental methods which have been used to measure the de-

formabilities of single micro-cellular entities. This includes the nano-/ micro-

indentation methods, the micropipette aspiration technique and the compres-

sion method. The second part describes the current experimental technique;

the micro-upsetting method. Finally, a conventional Instron Machine used to

compress macroscopic materials, in order to study the scaling effect for the de-

formation, is also introduced.

2.1 Review of the previous methods

There are various experimental techniques which allow the characterisation of

the deformabilities of single micro-cellular entities. The fundamental principles

of these different techniques are briefly described, and the relative merits and

demerits of each method are also discussed in detail in the following sub-sections.

2.1.1 Nano-/Micro-indentation

The indentation hardness test, where a conical or pyramidal indenter creates a

localised deformation in solid material, is a relatively simple and virtually non-

30

2.1 Review of the previous methods 31

destructive approach for characterising several mechanical properties of solids.

By adopting appropriate analyses, the method can provide useful information

about the spatial variations of these properties in nonhomogeneous samples.

These features make the indentation test a feasible method to estimate material

properties. There are two different types of methods which are conventionally

used to measure indentation hardness; the imaging method and the compliance

method (see Figure 2.1). In the imaging method, on the test surface an indent

is created by an indenter which is imposed by a specified load and withdrawn

after a certain dwell time. The hardness is then calculated from the values of

the applied load and the computed area of contact, which is estimated from the

diameter or diagonal dimension of the indentation image. It is especially well

suited to metals for example which deform plastically. However, this method

has been shown to be limited for organic polymers which creep and relax after

unloading. The compliance method measures the reaction force on the indenter

as a function of the depth of penetration, resulting in a set of loading and

unloading curves for each indentation. The compliance curve with appropriate

analysis provides information on mechanical properties for all classes of materials

(Sebastian 1994).

Recently Daily et al. (1984) and Zahalak et al. (1990) have developed a

so-called "cell-poking" method which involves indenting animal cells (ca 10 pm)

with a micro-radius tip (ca 2 pm) to measure the loading and unloading response.

This method may be considered as an extension of the compliance method.

However, for a number of important reasons the present work has preferred to

use micro-platens (see section 2.2) to compress or upset micro-cellular entities

rather than to use micro-indentation method. Firstly, micro-indentation can

provide useful localised mechanical properties only if the indenter tip size is

small (by a factor of ten) compared with the size of the single cellular entities.


This requirement seems not be fulfilled in the group experiments described by

Daily et al. (1984) and Zahalak et al. (1990). Furthermore, most of the previous

works on polymer films using indentation have shown that when the depth of

indentation exceeds that of 10 % of the film thickness, the material properties of

film substrate may also contribute to the response of the indentation. This

intractable problem may also occur when such small particles are deformed.

More seriously, the observed deformation characteristics will be a very strong,

and also unknown, function of the precise geometry and size of the indenter.

These factors make the previous analysis of the data extremely difficult and the

value of the results uncertain.

2.1.2 Micropipette aspiration method

This method has been widely used to study the mechanical properties of single

liquid-filled membrane systems including both microcapsules and biological cells

(see Figure 2.2). However, up to now the method has not yet been applied to

homogenous elastomeric spheres. In this technique, a portion of the membrane

of the cellular entities is drawn into a pipette by applying a pressure difference

between the pipette interior and the ambient media. The measurement of the

interrelationship between the length of the aspirated projection of the membrane,

L, and the applied pressure difference, zP, combined with an appropriate

analyses, provides the useful information about the mechanical/ viscoelastic

properties of the membrane. However, there are some serious disadvantages

existing in the application of this method. First, this method is only suitable

for membranes undergoing small deformations. At large deformations of the

membrane the adhesive or frictional force between the inside of micropipette and

the membrane of cellular entities will be a significant, but of unknown extent.

A more serious problem is that the nearly infinite curvature located in the tip

P N>7

P

2.1 Review of the previous methods

33

(a) Imaging method

L____________________

(b) Compliance method

Figure 2.1: The two methods of indentation hardness measurement: (a) the

imaging method, when the diameter or diagonal, d, is measured after indenta-

tion; and (b) the compliance method, where the reaction force, P, and the depth

of penetration, h, are measured continuously during the indentation. Adopted

from Sebastian (1994), with permission.


of aspirated projection easily causes the cell to be damaged. In addition, fortè-

biological cells the contributions of some of the intracellular organellest to the

observed mechanical properties of the whole cell has been shown to be significant

when using the micropipette method. Essentially, the potential intractability of

quantifying these above effects makes the micropipette method such that it is

difficult to obtain unequivocal mechanical and viscoelastic properties.

2.1.3 Compression method

The compression method, which involves the squeezing of individual micro-

cellular entities between two parallel platens, allows for the mechanical and

viscoelastic properties to be determined in a relatively unequivocal manner (see

Figure 2.3). This method has been widely applied for various materials. Cole

(1932) compressed sea-urchin egg cells (ca 120 jim) with known forces and from

measurements on the deformed geometry calculated the tension in the mem-

brane. Recently Zhang et al. (1992), using the compression method, have de-

veloped a micromanipulation technique to squeeze single cells (ca 15 tim) , and

to measure simultaneously the force being imposed on a cell and its deformation.

They applied this technique, to both mammalian cells and microcapsules, and

combined with appropriate analyses, the tension modulus and bursting strength

can be determined (Zhang et a!. 1994). Shipway and Hutchings (1993) have

presented a theoretical and experimental study of the fracture of single brit-

tle spheres (ca 750 jim) by uniaxial compression between opposed platens and

by free impact against targets. However, the central problem with the previ-

ous applications of the compression method is that the requirement of a high

position resolution of the platen movement has been proven to be difficult to

The internal volume of the cell, exclusive of the nucleus, is occupied by membrane-bounded

compartments called organelles.

L\P

I


Figure 2.2: Micropipette aspiration method: LP is the pressure difference be-

tween the pipette interior and the outside medium, and L is the length of the

aspirated projection of the membrane.

2.2 Micro-upsetting method 36

achieve. Furthermore, there have been no experimental techniques which permit

the measurement of a continuous force-displacement curve for both the loading

and unloading processes.

2.2 Micro-upsetting method

Based upon the compression method, the microupsetting method which allows

the compression of single cellular entities between two parallel micro-platens has

been developed during the current study. Unlike the conventional compression

method, this method, which is essentially an extension of the compliance method

adopted for nano-/ micro-indentation, can continuously measure the loading/

unloading cycles. In order to precisely measure the load-displacement curve, the

micro-platen is driven by a micro-stepper motor so as to move at various speeds

with a sufficiently high degree of accuracy (about 100 nm). Furthermore, an

incorporated visualisation component allows for the investigation of the corre-

sponding major geometric deformational parameters. Thus, the instrument has

essentially the same elements as the imaging method for indentation.

2.2.1 Equipmental setup

The primary functions of the instrument were to provide a capability to simulta-

neously measure both the approach and the resultant forces, whilst compressing

a small single sphere between parallel glass platens in either a dry or fully solvent

swollen state. Optical viewing of the deforming particle in a vertical plane pro-

vided a reasonable estimate of the central lateral extension of the particle and a

much less accurate measurement of the contact area. The instrument system is

schematically shown in Figure 2.4 and is based around an inverted optical micro-

scope (Wilovert S, Wetzlar Ltd., Germany). Attached to the microscope stage

(Z plane) was a microstepper motor controlled motion stage (PTS1000, Photon

IiF


37

Apply displacement

Measure Load

Visualise event in two planes

F

X7

Figure 2.3: The compression method: F is the force on each plate where the

plates are separated by each other a distance Z.


Control Ltd., England) capable of discrete micro-steps of less than 100 nm. At-

tached to this vertically (Z axis) orientated motion stage was a small horizontal

arm on which was mounted a very sensitive force transducer (BG-10, Kulite

Ltd., USA) with a force resolution better than iO N and a maximum force

capability of 10_i N. The instrument developed had a force and a displacement

resolution of 10 jiN and 0.1 jim, respectively.

On the lower face of this transducer arm was attached a small flat glass

platen (0.7 mm diameter; ca 100 im in thickness) for deforming the sample. The

force transducer signal was amplified and filtered using a strain gauge amplifier

(369TA, Fyde Ltd., U.K.). The absolute position of the platen was monitored

with an optically encoded displacement transducer (MT25B, Heidenhain Ltd.,

Germany) which resolved a vertical displacement of 100 nm over a 25 mm range.

The entire instrument system was computer controlled by using a purpose

written software. Output from the strain gauge amplifier was converted from an

analogue to a digital signal using a 12 bit analogue to digital card (PC LPM 16,

National Instrument, USA) installed within an IBM compatible 486 computer

(see Figure 2.5). The software developed allowed the complete control and

monitoring of the measured force, as well as the displacement, the velocity and

the acceleration of the stage and thus the platen. Two main modes of instrument

operation were used. The first mode was simply the micro-cellular entities being

loading followed by an unloading for a predetermined total strain or displacement

at a nominally constant velocity. The second mode of operation allows the

force to be measured as a function of time for a constant initially imposed

displacement and then following the reaction stress of the sample; essentially a

stress relaxation experiment.

During all the experiments a high resolution video camera (TM 620, Pulnix

Ltd., USA) was connected with the microscope in order to measure the sphere's


A ParticleB Petri DishC Indentor (attached a

small glass platen)D Force Transducer I GEArmF Translational StageG Position SensorH Visualisation PositionI Microscope StageJ Cantilever Beam E

VFigure 2.4: The schematic view of the micro-upsetting instrument (not to scale).

Strain GaugeAmplifier

InterpolationElectronics

MicrostepDrive Unit

A-D AcquisitionCard HOST

COMPUTER

Stepper MotorController

2.2 Micro-u psetting method

40

Force Transducer Optical EncoderPosition Sensor

Motion Stage andStepper Motor

ISO BUS RS-232 ISO BUS

Figure 2.5: The schematic view of the computer-controlled systems of the micro-

upsetting instrument (not to scale). The items outside the boxed region marked

as "Host computer" were mounted upon a vibration isolation table.


diameter both before and during the experiment. The initial calibration of this

optical system was carried out using a micrometer which was laid on the petri

dish and measuring the scale of its corresponding magnified image obtained from

the video camera.

The whole mechanical and optical components of the instrument system, ex-

cluding the power supplier and data acquisition was mounted upon a pneumatic

vibration isolation table (A2 LRS, Photo control, Cambridge, U.K.). Extensive

electrical shielding was required to suppress electrical interference in the data

acquisition leads.

Under large loads the cantilever beam of the force transducer may deflect

sufficiently to cause a significant difference between the sensed imposed displace-

ment and the actual imposed displacement. This difference is represented as the

quantity LIX in Figure 2.6. Routine calibration t of the instruments compliance,

principally that of the force transducer, allows the extent of this difference to be

accurately determined. Hence, all the experimental load-displacement curves in

the current study have been corrected for the deflection of the force transducer.

Typically, the maximum extent of this correction to the displacement is ca 18

tim, the spring constant is ca 1.67x iO N m 1 . The potential error estimated

without this correction is about 15 % in the displacement throughout the range

for this source.

In order to examine the parallelism of the plates, a photograph of the ar-

rangement of the indenter and the petri dish was taken from the side view

(shown in Figure 2.7(a)). The maximum error in the measured displacement

tThe deflection distances of cantilever beam were measured by an optically encoder

(MT25B, Heidenhain, Ltd., Germany), as a function of the imposed forces (without the sam-

ple). Hence the actual imposed displacement of the deformed cellular entity, under a certain

force, may be obtained by deducing the corresponding deflection distance from the sensedimposed displacement.

Platei

er


42

AX Deflection distance

L Length of cantilever beam

0 Rotation angle

Figure 2.6: The deflection of cantilever beam under large imposed loads (not to

scale). X is the maximum deflection distance and 0 is the rotation angle of

micro-platen caused by the deflection.


43

-.,:9

(a)

L

(b)

Figure 2.7: (a) A photograph of the arrangement of two parallel platens: the

indenter (micro-platen) and the petri dish (see the following diagram for details).(b) An idealised schematic view of the platens of the micro-upsetting instrument

(not to scale). LY is the maximum error in the measured displacement caused

by the imperfect parallelism of these two platens, and dy is the real error arising

from this source. L is the diameter of the indenter and D is the diameter of the

contact area between the deformed cellular entity and the platens.


which was caused by the initial imperfect parallelism of these plates is described

as the parameter LxY in Figure 2.7(b). This error was measured to be approx-

imately 20 tim. However, the real error arising from this source may expressed

as the quantity dy in Figure 2.7(b). if the surface of the platen is assumed to

be perfectly flat, a linear relationship between LY and dy may be simply given

by;LYD

dy= L

where L is the diameter of the micro-platen (about 0.7 mm in the current case)

and D is the diameter of the contact area between the deformed cellular entity

and the plates. In the current study, for a ca 300 /Lm elastomeric sphere, at

40 % deformation, the radius of the contact area is approximately 110 1um.

Then the real error dy is calculated as about 3 um which is equivalent to ca

1 % deformation. At large imposed loads, due to the corresponding cantilever

deflections, the initial parallelism of the platens was distorted; see Figure 2.6.

This distortion may be idealised as a rotation in the micro-platen by an angle, 0.

Hence, the measured forces sensed by the force transducer, which is less than the

reaction forces of the deformed entities, F, may be equivalent to Fcos0 (again,

see Figure 2.6). The maximum consequent error in the force measurement is

estimated as about 1 % in the current case. Under no circumstances did any

detectable translation of the particles occur.

2.2.2 The major merits of the current method

There are some major advantages of the current method, compared with others,

which are listed below;

1. The experimental data which are obtained by using micro-platen compres-

sion (micro-upsetting) method are much more amenable to analysis as the

extent and geometry of deformation is more exactly defined.

(2.1)


2. It seems that the micro-platen upsetting technique will avoid the possible

damage of the cellular systems upto quite large deformations.

3. It allows the measurement of elastic and viscoelastic time-dependent defor-

mations; for example, it allows a stress-relaxation experiment to be per-

formed.

4. The method gives a continuous force-displacement curve for both the loading

and unloading.

5. The method may be universally applied to both the homogenous elastomeric

spheres and the liquid-filled membrane systems.

6. The method can estimate the bursting strength of the liquid-filled membrane

systems.

Finally, apart from the above advantages, a striking feature of the current

experimental approach is that it allows the assessment of the autoadhesive ef -

fects between the elastomeric spheres and the micro-platens by measuring the

"pull-off" force (see Chapter 4). The magnitude of the interfacial free energy can

be estimated by adopting an appropriate analysis. In fact, a similar apparatus

has been used by Ducker et al. (1991) to study the interaction force between a

single silica sphere (ca 3.5 m) and a silicon surface. Essentially their apparatus

was a atomic force microscope (AFM) on which a V-shaped tip was mounted, as

a cantilever. A particle was attached at the end of tip and the tip approached a

planar surface. The interaction force between the particle and the surface causes

the deflection of the cantilever. Then, a signal, proportional to the deflection as

function of the distance between the particle and the surface, may be produced.

Hence, by converting the data on the deflection into displacement (if the spring

2.3 Instron Universal Testing Machine 46

constant of the cantilever is known), the force-displacement curve can be ob-

tained. Their apparatus as described had a force and displacement resolution of

0.2 nN and 0.3 urn, respectively. However, it is obvious that, although the force

and the displacement resolutions between the current instrument and Ducker's

are different, the general principle may be considered to be the same.

2.3 Instron Universal Testing Machine

In order to explore the effects of size scales on the mechanical response of elas-

torneric materials, macroscopic elastomeric spheres were studied, in addition to

the microscopic particles, using an Instron Machine. The Instron Model "6022"

has been employed in the current study to compress poly(urethane) spheres (ca

38.3 mm) under a constant applied strain rate (or loading speed). The basic

instrument consists of a loading frame and a series 600 digital control console.

The constant strain rate control was achieved with an IBM compatible personal

computer fitted with an IEEE interface card (CIL Group). The applied strain

rate was monitored and corrected by the computer-controlled system at 0.5 sec-

onds intervals. The values of the load, the compressive displacement and the

applied strain rate were collected through a "data acquisition" system (PC LPM

16, National Instrument, USA) and recorded in the computer.

The compression load range was between 0 to 10 kN, the applied strain rate

was adjusted between 0.05 to 1000 mm/mm.

Chapter 3

Theoretical models

This Chapter reviews some of the theories available for the description of the

deformation of both homogenous solid spheres and liquid-filled membrane sys-

tems. For the compression of the homogenous solid spheres, the theories with-

out the inclusion of the autoadhesive effect, both the Hertz (Hertz 1882) and

Tatara theories (Tatara 1991), as well as a theory with the autoadhesive effect,

the Johnson-Kendall-Roberts (JKR) theory (Johnson et al. 1971), are described

briefly. For the liquid-filled membrane systems, the first-order theory, the liquid-

drop theory, and the high-order theories such as, the membrane model (Feng &

Yang 1973) as well as the shell model (Taber 1982), are reviewed. Finally, in this

Chapter a systematic approach to identify the choice of appropriate theories for

the various structures and the different stages which may occur in the loading

and the unloading processes is also introduced.

3.1 Review of the theories

There are a number of classic theories which have been proposed, or applied, by

various authors for both the linear and non-linear compressive deformation of

homogeneous solid spheres, as well as liquid-filled spherical membrane systems,

at both small and large imposed strains (see Table 3.1). The corresponding

47

3.1 Review of the theories 48

experimental examination of these theories is also summarised briefly into the

same Table.

The Hertz theory, a well-known theory in contact mechanics, describes the

small strain deformation of spheres in an elastic range in the case of a nor-

mal, nonconformal, and frictionless contact of two solid bodies with a similar

Young's modulus and Poisson's Ratio. The JKR theory modifies the Hertz the-

ory by including an autoadhesive effect which plays an important role in the

very small deformation range. The Tatara theory invokes non-linear elasticity

and a large deformation formulation for predicting the compressive behaviour of

elastomeric spheres at large deformations. For the liquid-filled membrane sys-

tems, the liquid-drop model considers that the membrane behaves essentially as

a soap film with the surface tension forces acting uniformly in the two principal

directions on the surface and not changing with the extent of the compression.

However, it is obvious that this model is not appropriate for the elastic mem-

brane systems, such as the liquid-filled microcapsule with a polymeric membrane

which is used in the present study. The elastic membrane model treats these

systems as a thin-walled non-linear elastic membrane in which the enclosed vol-

ume remains constant under compression and that the membrane stretching,

but not the bending, forces govern the deformation. However, the shell model

describes the system as a thick-walled member in which both the stretching force

and the bending moments govern the deformation. For simplicity, the details of

these theories, which includes their basic assumptions, final equations, as well

as relative merits and demerits are presented elsewhere in this Thesis. The

JKR, Hertz and Tatara theories, are discussed in detail in Chapter 4, whilst the

membrane model and the shell model, are reviewed in Chapter 5. In addition,

the Lifshitz theory (Lifshitz 1956) which is potentially applicable to describe the

"jump contact" phenomenon (Tabor & Winterton 1969) between the spheres and

3.1 Review of the theories 49

Table 3.1 Various models available for the prediction of the deformation of

micro-cellular entities under Compression and Indentation

Authors

Tabor and Win terton

(1969)

Johnson eta!. (1971)

Models

Lifshitz theory (1937)

Experiments

Jump contact between two

cylindrical sheets of mica

JKR theory (autoadhesive Contact of rubber &

effect) gelatine spheres

Yoneda(1973) Liquid-drop model Compression of sea urchin

(the uniform tension on the eggs between the platens

surface of cell)

Lardner and

Pujara (1980)

Taber (1982)

Zahalak et a! (1990)

Zhang et al (1993)

Membrane model

(stretching energy)

Shell model (bending &

stretching energies)

(1) Liquid-drop model

(2) Finite element analysis

for Hookean material*

Liquid-Drop model

None

Compression of fluid-

filled spherical shell by a

rigid indenter

Indentation of leukocytes

Compression of

mammalian cells

Tatara, Tatara et a!.

(1991,1993)

Shima et al.(1993)

(1) Hertz theory (linear

elasticity & small

deformation)

(2) Tartar theory (nonlinear4elasticity & large

deformation)

Compression of

homogenous rubber ball

* This is outside the scope of the current study.

3.2 The identification of applicable theories 50

the platen, attached on a cantilever beam, is described in Appendix A along with

an outline of a new interpretation of the "jump phenomenon" provided by Prof.

K. L. Johnson.

3.2 The identification of applicable theories

It is obvious that most of these theories described above, corresponding to differ-

ent mechanisms, were originally developed for application to specific systems.

Henc, an interesting question may come out as this: "Can these systems be

distinguished according to the differences in their mechanical deformation be-

haviours?". To answer this question, it may be useful first to examine the load-

ing/ unloading curves for the different systems. The Figure 3.1 shows the general

features of the loading! unloading curves for three different types of structures;

homogeneous elastomeric spheres, liquid-filled spherical elastomeric membranes

and spherical shells. For a homogeneous elastomeric sphere, the loading and

unloading curves are nearly identical. The yield stress of elastomer is assumed

to be extremely difficult to achieve; that is the sphere does not fracture under

the maximum loading force. However, for the liquid-filled membrane system, the

reaction forces for unloading sharply drop to the zero after membrane bursts.

For the liquid-filled elastic shell, the reflex force, which is contributed by the

bending moment, clearly shows in the loading curve after shell ruptures.

Regarding the deformation for homogeneous elastomeric spheres, there are

four major theories. These are the Lifshitz, JKR, Hertz and Tatara theories,

which are available for the predictions of the different stages which may occur

sequentially in the loading! unloading operations (see Figure 3.2). Of course, all

of these four theories should be considered as the descriptions of diverse mecha-

nisms which govern the deformation responses. The Lifshitz theory is applicable

4.)U

0

3.2 The identification of appiicable theories

51

Displacement

Figure 3.1: The general features of the loading! unloading curves for three

different types of structures: (a) homogeneous elastomeric sphere; fracture stress

is assumed to be extremely difficult to achieve; that is the sphere does not

fracture under the maximum loading force. (b) liquid-filled spherical elastomeric

membrane; (c) liquid-filled spherical elastomeric shell when wall rupture occurs.


for the description of the "jump contact" which is mainly caused by the balance

between the potential autoadhesive force and the bending force of the cantilever

beam. After the sphere contacts with the platen, then the JKR theory may

be applicable to interpret both the loading and unloading responses which are

mainly governed by the autoadhesion and contact elastic stresses, at small de-

formations. During the unloading process, the "pull-off" phenomenon, which

is a negative (tensile) force created by the autoadhesion force when the platen

separates from the spheres may be uniquely predicted by the JKR theory. When

the deformation (dimensionless approach*) is approximately upto 5 %, the Hertz

or JKR theory, depending upon the extent of the contribution from the autoad-

hesion, may be applied to describe the loading and unloading responses. Once

the deformation is larger than about 10 %, the Tatara theory may be consid-

ered as the most suitable model for the interpretation of the force-displacement

curve. Therefore, we may conclude that, in principle, through the characteri-

sation of the compliance curves, for both the loading and unloading processes,

the appropriate theories for the systems may be identified. Thus, in summary,

a careful examination of the features of the loading and unloading curves may

provide an indication of the most appropriate analyses or theories. However,

these judgements are more securely confirmed by a corresponding examination

of the geometry of the deformation.

Dimension1ess approach is defined as the compressive displacement normalised by theinitial particle diameter.

4.,U0

I, •S lI


ci)

I,, t

/ S I I /1,

(a) Lifshitz theory (b) JKR theory (c) Hertz theory (d) Tatara theory

Figure 3.2: The various theories available for the descriptions of the sequential

stages, appearing in the loading and unloading, for the elastomeric micro-spheres

Chapter 4

The deformation of

homogeneous micro-elastomeric

spheres

This Chapter reports upon a theoretical and experimental study of the com-

pressive behaviour of microscopic (ca 300 jim) elastomer spheres over a wide

range of imposed deformations. The experimental results presented for these

micro-elastomeric spheres confirm the theoretical predictions of various limiting

case models for the dependence of the reaction force on the compressive de-

formation of an homogeneous sphere. At values of the dimensionless approach

(compressive displacement /initial particle diameter) up to 10%, the classical

Hertz theory was found to be in good agreement with experimental results and

confirms that the load is a function of the approach to the 3/2 power. At

larger deformations (dimensionless approaches from 10% to 37%), a numerical

implementation of Tatara's large deformation model for the compression of an

elastomeric sphere (Tatara 1991) gives good agreement with the experimental

results. The Tatara analysis provides a numerical solution in which the load de-

pends upon the approach to the cubic power for large deformations, and follows

the fifth power of the approach for even larger deformations. However, these two

54

4.1 Introduction 55

theories, the Hertz and Tatara theories, both neglect the autoadhesion effects

which are important in the small deformation region for certain cases. At zero,

or very small, deformations (dimensionless approach < 1.0%), an established

autoadhesion theory, the JKR theory (Johnson et al. 1971), accurately predicts

a pull-off force which is independent of the dimensionless approach. This predic-

tion has been confirmed experimentally. The analysis also accurately predicts

the form of the compliance curve.

4.1 Introduction

For a non-adhesive elastic sphere compressed between two parallel flat platens,

the force resisting deformation depends upon the approach to the 3/2-power for

small deformations. The theoretical nature of this relationship was described

in detail by Hertz (1882) and allows the deformation of the sphere in the re-

gion of the contacting platens to be fully described subject to a number of

important assumptions. The principal assumptions are that a normally loaded

non-conforming contact exists between the bodies; the material behaves as a

linear elastic body; the radius of contact area is small compared witliadius ofthe

the sphere; and that there is a frictionless contact between the surfaces result-

ing in the transfer of only normal stresses between the contacting surfaces. The

success of this model has been established, for large contacts, by a number of au-

thors, as have the limitations inherent in this theory. Some important extensions

proposed for Hertz's theory, especially for the pressure (normal stress) distribu-

tion in the contact area, have been given by Johnson (1985). Also the modified

stress (including normal and tangential stresses) distributions for rolling con-

tacts have been determined by using numerical methods (Bentall & Johnson

1967). For large deformations, Yoffe (1984) proposed a modified Hertz theory

4.1 Introduction 56

which involved a first-order correction for the errors which are introduced when

relatively large contact areas are present. Apart from purely elastic deformation,

Johnson (1968) has included a modified pressure distribution to allow for the

presence of plastic deformation within the contact region. This refinement is not

however relevant to the description of the ela.stomeric materials which are the

subject of the current study. The influence of interface friction was considered

by Johnson et al. (1973).

In a detailed consideration of the compressive mechanics of microscopic single

spherical particles, a number of specific phenomena need to be considered which

are not generally incorporated in the original Hertz theory. Specifically, these

are the influence of the adhesive surface forces and the effects of large strains. In

the case of the former, the mechanical behaviour of microscopic particles at low

loads is dominated by the action of surface forces, especially for softer materials.

Whilst for these same soft organic materials, the low strain limitation of Hertz

may easily be exceeded in many applications.

For the other loading estimate, in the case of a non-adhesive elastic sphere

undergoing a large deformation, recent experimental and theoretical work by

(Tatara 1991; Tatara et al. 1991) has shown that the elastic force is propor-

tional to the approach raised to the third power at large deformations (20 %

dimensionless approach). At very large deformations, the load dependency fol-

lows the fifth power of approach. Tatara's theory invokes non-linear elasticity

and a large deformation formulation for investigating the compressive behaviour

of elastomeric spheres upto a value of 60 % of the dimensionless approach. The

influence of autoadhesion is not considered nor are the effects of interfacial fric-

tion.

Work by Johnson, Kendall and Roberts; "JKR theory" (1971), has demon-

strated the importance of including the van der Waals' adhesive forces (or indeed

4.1 Introduction 57

any adhesive force) in describing the contact mechanical behaviour of smooth

spherical low modulus elastomeric bodies at very low loads, including zero ap-

plied load, and have detailed the modifications of the Hertz theory necessary

in order to consider these additional surface forces. In their work the adhesion

force between two solid bodies was assumed to operate over an infinitely short

distance. Contrary to the Hertz theory which assumes only compressive stress in

the contact area, the JKR theory allows for the contributions of tensile stresses as

well. In fact, JKR theory adopts an infinite tensile stress at the edge of the con-

tact circle where the surfaces are expected to deform infinitely sharply through

an angle of 900. However, Derjaguin, Muller and Toporov; "DMT theory" (Der-

jaguin et al. 1975) later developed an alternate theory which was essentially

based on the argument that the adhesion force between the solids must operate

over a finite distance and thus operates in the region just outside the contact

zone where the surfaces are a small distance apart. Hence, unlike the JKR ap-

proximation, the DMT theory assumes that the influence of attractive forces on

the stress distribution inside the contact zone are negligible. Furthermore, the

original DMT theory assumes that the deformed axial plane profile shape of the

surfaces is not affected by the surface forces and is thus Hertzian in its behaviour

whereas JKR theory assumes the resultant profile to be non-Hertzian. Although

different assumptions are inherent in both of these two theories, they both recog-

nise that the autoadhesion effect plays an important role in the deformation of

adhesive soft elastomers at very low loads.

For distinguishing the transition from the DMT to the JKR theory (both

may be regarded as approximations) Tabor (1977) was the first to introduce

a single dimensionless parameter, q, comprising the radius of particle, its elas-

tic constants, and the characteristics of the potential of interaction with the

substrate (see section 4.2.2). Subsequently, other authors have proposed other

4.2 The theories of deformation 58

single parameters, such as either i by Muller et al. (1980; 1983) or .A by Maugis

(1992), which are proportional to the parameter suggested by Tabor (again see

Section 4.2.2). These authors have agreed that for << 1 (hard solids of small

radii and low surface energy) the DMT analysis provides a better description,

whilst for i >> 1 (low elastic modulus material with large surface energy and

radius) the JKR theory is a more accurate description.

Although experiment work on macroscopic bodies has separately verified the

general validity of the two preceding theories, the autoadhesion theories and the

Tatara analysis, the unified verification of these phenomena, on a microscopic

scale, has not been demonstrated prior to the present work. The current study

provides a direct experimental verification of the relevance of these theories

with respect to microscopic particles and utilises these analyses as the basis for

a first order mechanical and interfacial characterisation of soft elastic spherical

microscopic particles which include non-Hookean behaviour, surface adhesion

interactions and large strain deformations.

4.2 The theories of deformation

4.2.1 The theories without autoadhesion effects

4.2.1.1 Hertz theory

Hertz's theory has been experimentally shown to be valid at small deforma-

tions by compressing a non-adhesive elastic sphere between two rigid planar

substrates. When the elastic spherical body is subjected to an applied contact

normal load (P), contact will occur between the sphere and the rigid planar

substrates in a circular region of radius(a), given by the following equation

13(1 —z2)RPa

= L 4E

]1/3(4.1)


59

where R is the radius of the sphere, E is the Young's modulus and ii is

the Poisson Ratio. Moreover, the approach of a Hertzian contact (as ) can be

represented as:

3(1_u2) 2/3c2

= [ 4E/ Jp213 (4.2)

4.2.1.2 Tatara theory

Recently, Tatara (1991) has developed a general theory for the compressive

deformation of a homogeneous nonadhesive elastomeric sphere under large de-

formations; say greater than 15 % nominal transverse strain. Tatara's theory

may be considered as an extension of Hertz's theory which removes two of the

main assumptions inherent in this classic model; that is the limitation of small

imposed deformations and the requirement of a linear elasticity. In order to un-

derstand the Tatara's theory, a concise description of the modifications adopted

in his formula for the case of large deformation is given ijgl1owing paragraph.the

As described in the classic contact mechanics literature (Timoshenko &

Goodier 1970; Johnson 1985), the vertical compression displacement, w(z, r),

and radial displacement, u(z, r), of half-space elastic body 1 created by a con-

centrated force (F) acting along the z axis at the pole on the surface (see Figure

4.1) in the cylindrical coordinates (z,r) are given by

w(z,r) - P 1 (1+u)z 2 2(1 —u 2 ) 1 (4.3)- 2irE [(r2 + z2)3/2 + 2 + z2)1/2](r

= (1+u)P[ r2z I zu(z, r)

2ER L(r2 + z2)312 - ( 1 - 2u) \ (r 2 + z2)h/2)] (4.4)

The half of the compressive displacement at the pole of the deformed sphere.The half-space elastic body which is defined as a semi-infinite elastic solid bounded by

a plane surface. This idealisation, in which bodies of arbitrary surface profile are regardedas semi-infinite in extent and having a plane surface, is made almost universally in elastic

contact stress theory. It simplifies the boundary conditions and makes available the largebody of elasticity theory which has been developed for the elastic half-space (Johnson 1985).


where w(0,r) = (1 - v2 )P/2irEr corresponding to the Hertz's approach, c,

although Hertz (Hertz 1882) derived it from the electrostatic analogy. It is worth

remembering that the equations (4.3) and (4.4) were derived by Boussinesq

(Boussinesq 1885) under the following two conditions (apart from the condition

of linear elasticity):

(i) The displacements, u, v, w in each direction of x, y, z at a point are small

as noted by = ôu/öx (here is the strain in the x direction).

(ii) The lateral strains are noted by = —vc.

In addition, in most of the conventional analyses the following critical condition

is often assumed

(iii) The radius, a, of the contact area is small in comparison with the radius,

R, of the sphere (Timoshenko & Goodier 1970; Johnson 1985).

These three conditions which may be summarised as small strain, (i) & (ii), and

small displacement, (i) & (iii), are called the assumption of "small deformation".

In general, elastic bodies like ela.stomeric sphere may undergo large strains

while the derivative of strain at each point remains small; a strain () at a point

may be expressed by the strain () at the neighbouring point as the first order;

noting (Tatara 1991)

= + öf/DXdX (4.5)

where and are large while a&/ax is small. Hence, the condition (i) may

be valid in large strains and equations (4.3) and (4.4) can be extended to the

case of large deformations, with the following modifications. According to the

theory of rubber elasticity (Treloar 1970; Green &Adkins 1970), the lateral

extension ratios X,,=) =i/./X at constant volume of \X,AZ = 1 (where )

is the extension ratio of elastomeric sample), is different to the condition (ii):


= —v. In addition, the symmetry of the displacements and the strains of

elastic medium in the sphere due to the reaction at the opposite contact surface

at z=2R is considered. As a result, the vertical displacement W(u,r) and the

radial displacement U(z, r) at the point (z, r) in the sphere due to a concentrated

force, F, and the reaction, —F, may be represented by

W(z, r) = w(z, r) - w(2R - z, r) (4.6)

U(z, r) = u(z, r) + u(2R - z, r) (4.7)

where w(z, r) and u(u, r) are given according to equations (4.3) and (4.4). With

these above modifications, Tatara extended the half-space elastic body model to

the case of large deformation.

Furthermore, the Tatara model includes for the case of large deformations a

strain-dependent Young's modulus model (the Poisson's ratios is assumed to be

invariant) for the behaviour of the non-linear elastic material which is based on

the Mooney-Rivlin law for elastomer deformation. Therefore, Tatara modelled

the strain-dependent Young's modulus as a nonlinear function of the strain, f,

which can be expressed as below;

(1—f) (4.8)

where the strain is positive in compression, and E0 is the Young's modulus at

f0.

A significant advantage of this theory is that it allows for the full calculation

of the deformed shape of the sphere for large strains which cannot be calculated

by the Hertz Theory; the Hertz Theory does not address this problem only the

contact deformation. The details of the derivation of the necessary equations are

given elsewhere (Tatara 1991; Tatara et a!. 1991). The primary disadvantage

of the model (apart from the lack of a consideration of autoadhesion) is the


need for a numerical solution to the governing equations. The final equations

developed for this analysis are shown below:

(4.9)a = a + U (a,z)

z = R - /k2 - a2

PR - v'7— a2 = A (__3

(i - 2) (i -

+ Bf(a)a2)

8E0 a 2R2 2irE0R2

3 PA I' Ba2 " f(a')PA 1' Ba2'cr = (i - 2) (1 +

- irE0 (1 +

U(z, a) - A(1 + v)P i'

Ba2

- 2irE0+)

( (.4/i+V2R_z) (1_2u)(2\./_fi_\/2R_z))

2JR3/2 - J2Rz (2R - z)

(Ba2) (_(1_2u)(1_))

irRE0 1+:

(4.10)

(4.11)

(4.12)

(4.13)

(4.14)

= [Lf(H) - at Hf( L )]/[f( t H ) - f(atL)J f(H)f(L) <0 (4.15)

where()2

1- 3B='-e+c

1—v 2(2+2v)R2f(a) = _______ +

..1a2+4R2 (a2+4R2)31"2

(4.16)

(4.17)

(4.18)

(4.19)


63

z

Figure 4.1: Schematic of the symmetrical shape of a soft elastic sphere com-

pressed, at large stains, between two flat rigid surfaces, and the movement of apoint C(z, r) on the surface before deformation to C'(z', i") after deformation,where U(z, i-) and U(R) indicate the lateral extensions at the contact surface

and the central diameter, respectively. The is the approach; a' is the com-

puted radius of the contact surface at the vertical position (z); a is the radius

before deformation. (adapted from Tatara's original paper (1991))


where c is the compressive approach distance (which was expressed as -y in

Tatara's original paper); a' is the computed radius of the contact surface at the

vertical position (z) with a lateral extension U(z, a); a is the radius without the

lateral extension; U(R) is the lateral extension at the central diameter; E0 is

the initial Young's modulus (the Young's modulus at small deformations). A

schematic diagram which concisely describes the geometric interrelationships of

the above variables is shown in the Figure 4.1. Equation (4.15) is the basis of

the "two points and circulation method" which was used by Tatara et al. (1991)

for determining the solution of crtM, the medium point of , by the two points,

L and ag", the high point and low point of .

The algorithm developed in the present work and adopted here is an enhanced

version of the original algorithm as proposed by Tatara et al. (1991). This new

version is significantly more efficient in computing the required solution set.

The flow chart for this algorithm is shown in the Figure 4.2. The improvement

in efficiency, in terms of computation time is circa a factor of 5. A computer

program developed based upon this algorithm is reported in Appendix B.

4.2.2 The theories with autoadhesion effects

Two models have been widely adopted to correct for the behaviour of Hertzian

contacts in the presence of surface forces; the JKR and DMT theories mentioned

previously in Section 4.1. An experimental investigation of the contact area

between optically smooth rubber and gelatin spheres for very low loads found

that the areas of contact measured were significantly larger than expected based

on the predictions of the Hertz theory. The JKR theory was directly developed to

explain these observed differences. This theory takes into account the additional

deformation near the contact periphery resulting from the interaction of surface

forces operative at the interface. Essentially, these authors modified the Hertz

END

YESYES

Sub F 367(y)

a=O to SR

fL=S ub F45C1,aL)

aH=a+O.1R

1LH>0

eq. (9) for afM=Sub F45(y,aM)

( STARJ

9)

fL=Sub F367(y)

YHY°f=Sub F367('y141

L1H>0

eq. (9) for YfM=Sub F367()

IYH- Yu<105

YES

NOLM >0 -

t. M


65

YESi aH - aU <

eq. (6)eq. (3)

eq.(7)t'L M >0

LM I H=M ]aL = a ,4 an—a

IM I

______ (TURN

(Sub F45(iaJ

eq. (4)eq.(5)

RE1U9

Figure 4.2: The flow chart for the enhanced version of the algorithm for solving

the set of the equations of the Tatara theory used in the current study.


theory for two adhering, smooth, elastic spheres to include the thermodynamic

work of adhesion, (L-y). By using an excess surface free energy relationship

similar to the Dupré equation (Wu 1982) the thermodynamic work of adhesion

introduced by these authors may be expressed as below;

= 'Xi + 'X2 - 'Y12 (4.20)

where 'y is the excess surface or interface free energies; the subscripts denote

the surface 1 and 2 and the interface 12. The analysis predicts that a finite

pull-off force is required to separate the solids, and that this force depends only

upon the mutual radii of curvature and their surface energies of the solids, but

not upon their elastic moduli. Rather remarkably the force does not dependent

on the original area of contact (Greenwood & Johnson 1981). Johnson et al.

considered a balance between the surface work, the stored elastic strain and the

potential energy. The final relationship for the autoadhesion (pull-off) force is:

PC = —3/2irRzy (4.21)

where P is the pull-off force or the force of attraction, R, is the mutual radius

of curvature. Generally, the mutual radius of curvature is:

1 1 1(4.22)

where R1 and R2 are the Radii of bodies. In the present case (flat plates in

contact with a sphere; 112 tends to infinity) R is equal to the radius of the

sphere.

Since the JKR theory takes into account the adhesion force of the solids,

by including the surface energy, the apparent Hertz load P1 acting between two

elastic bodies is greater than the applied load P. The relationship between P1

and P has been given by these authors as below:

Pi = P + 3Li'yirR + \I61rL'XRP + (3irz-yR) 2 (4.23)


Hence, equation (4.1) of the Hertz theory may be modified by including surface

forces which allows the radius of contact area (a3 kr) to be represented as below:

13R(1 -a3kr = 4E j

(4.24)

When Li-y is zero, the equation (4.24) is reverted to the equation (4.1) as pro-

posed by the Hertz theory. Furthermore, the approach can be expressed as

below:1jkr - [2yajkr (1 -

v2)Jh/2

(4.25)R E

where is the approach distance which is created by the compressive force

combined with the autoadhesion force.

In addition, another useful equation gives the pressure, or stress, distribution

within the contact circle as

_________________ _________________ 2 -1/2___________ 2.h/2 ( __2ELy 1/2

(1_x)- (12)) (1_x) (4.26)

8E7r

where x = r/a, k (see Figure 4.3). This equation indicates that a infinite tensile

stress occurs at the edge of the contact circle (at x=1), equivalent to a crack at

this region.

Detailed descriptions of the DMT theory are given elsewhere (Derjaguin et al.

1975). The theory may be concisely summarised by one principal equation, the

final relationship for the autoadhesion force, and is expressed as:

P = — 2irRL%'y (4.27)

The differences between the two approaches and their relative merits have

been discussed in detail by Tabor (1980), Derjaguin et al. (1980) and Maugis

(1992). They are both limiting solutions and there is a transition from the

DMT limit to the JKR limit which has been described by the increase in the

singk dimensionless system characteristic parameters which were suggested by


Equilibrium

(a)

Pull-off

(b)

Figure 4.3: (a) Deformable (elastic) spheres on a rigid surface in the absence

(Hertz) and presence (JKR) of adhesion. (b) elastic adhering sphere about to

separate spontaneously from an adhesive contact. Adapted from Israelachvili

(1991), by permission.


Tabor (1977), Muller et al. (1980) and Maugis (1992). Since all of these various

parameters suggested by different groups are proportional to each other, the

current study has simply adopted the form introduced by Muller et al. and it

may be expressed as below:

32 12Ry2 (1 - u2 2 1/3

=L irE2e3 }

(4.28)

where e is the equilibrium separation distance between the atoms at the interface.

The choice of the value of € is somewhat arbitrary; conventionally the value is

taken as the order of an interatomic distance. The transition region is in the

range of the values of of the order of unity. Values of u greater than unity are

consistent with systems which best approximate to JKR limit and vice versa.

For the current cases described later (R 150 m, iy 82 mJ m 2 , ii

0.48, E 2.06 MPa and e 5 A°), the i value may be calculated as the order of

1000. Hence, on this basis, the JKR theory is the most suitable approximation

for the current systems and it is adopted in the subsequent analyses described

in this Chapter.

The influence of surface topography and bulk viscoelastic contributions on

the response of these contacts has also been studied; for example, Fuller & Tabor

(1975), Johnson (1975) and Briggs & Briscoe (1977) have studied the reduction

of adhesion due to roughness. Various authors have noted the increase in ad-

hesive force due to viscoelastic contributions to the peeling work; Greenwood &

Johnson (1981), Andrews et al. (1982), Maugis (1985), Kendall (1987), Briscoe

& Panesar (1991) and Kendall (1994).

4.3 Experimental

70

4.3 Experimental

4.3.1 Material preparation

Poly(urethane) spheres were prepared from a commercial two part resin system

and were polymerised using the manufacturer specified cure cycle (Kemina Poly-

mers, UK, Diprame 54 series). Firstly, the two part resin system was mixed and

degassed under low vacuum. A single aramid ("Kevlar"49, Dupont de Nemours,

USA) monofilament, 11 micron in diameter and 25 mm long, was firmly attached

and stretched across the top of a fibre holder made from a U section of glass

rod. By carefully pulling the fibre holder through the surface of the resin system,

small droplets of resin would naturally form along the fibre length due to the

action of wetting forces. These large resin droplets, between 50 micron and 500

micron in diameter, were attached to the fibre in the glass holder and were then

placed in an oven at 60°C for 12 hours to allow the polymer to cure. The cured

microspheres were then carefully removed from the fibres . Optical examination

confirmed the spherical nature of the elastomer particles produced.

4.3.2 Experimental procedure

A particular micro-elastomeric sphere (ca 300 pm) was placed on a glass plate

situated underneath the microscope lens. Then, the original (undeformed) di-

ameter of the sphere was measured through the image of its bottom view shown

on the television linked with the microscope; see section 2.1.3. The microplaten

was moved down to the position of approximately 10 pm above the selected

sphere and then driven at a constant speed, typically 1.0 or 5.0 pm sec 1 , to

A hole was created by the removal of the fibre in the examined poly(urethane) sphere.

The volume of this hole is only a small fraction of the total volume of the sphere (about 0.24%). Hence, it may be reasonable to assume that the influence of this hole on the deformation

of the sphere was negligible.

4.3 Experimental 71

compress the sphere. The imposed force and the compressive displacement of

the deformed sphere during the loading and unloading (including the regions

of the "jump contact" and the "pull-off" stages) were automatically recorded

through the data acquisition system linked with the personal computer. The

diametrical extension of the deformed shape of the sphere was also continuously

measured and recorded by the microscope-linked video.

In order to explore the scaling effect on the mechanical response of these

materials, the deformation of a macroscopic elastomeric sphere (ca 38.3 mm)

was examined by compressing it between two parallel platens. The sphere was

manufactured in a simple model using the preparation method described for the

microspheres. One of these two platens, which is firmly attached on the "load

cell" of the Instron testing machine, is able to vertically move at a constant

speed and to continuously sense the reaction force of the compressed sphere.

The speed of the moving plate (or applied stain rate) was about 0.1 mm per sec-

ond, during both the loading and the unloading processes. The imposed forces

and the compressive displacements of the squeezed sphere were automatically

recorded by the data acquisition system associated with the personal computer.

However, the measurement of the diametrical extension of the deformed sphere

was carried out by another independent experiment. In that experiment, the

imposed deformation was gradually increased by a constant amount of displace-

ment (about 3.83 mm; equivalent to 10 % deformation). Following each increase

of the imposed deformation, the extended diameter of the compressed sphere was

routinely measured by using a pair of callipers.

The above experiments were conducted in the temperature range from 20 to

23 °C in ambient conditions; relative humidity ca 40%.

4.4 Results and discussions 72

4.4 Results and discussions

The experimental results, associated analyses and preliminary discussion are

provided in two main parts; Section 4.4.1 and 4.4.2. Section 4.4.1 discusses the

low strain (small dimensionless approach) data where the influence of autoad-

hesion predominates. Section 4.4.2, which is composed of three sub-sections,

considers the case of larger deformations where the influence of autoadhesion

may be neglected and the bulk mechanical responses will be the prevailing in-

fluence.

4.4.1 Autoadhesion effects

At zero and very low imposed loads (the dimensionless approach <1.0%) the

importance of surface adhesive forces is clearly evident in the experimental data

shown in Figure 4.4. This curve is a record of the imposed displacement and

the sensed (uncorrected) force provided by the load transducer for a 300 um

poly(urethane) sphere during a loading and unloading cycle. Between the points

S and A indicated in Figure 4.4 the upper platen (attached to the force beam)

approaches the sphere. At A the beam "jumps" into contact and a tensile force is

detected. Between A and B the sphere is compressively loaded; at A' the applied

load is zero. Unloading commences at point B and the following curve between

the points B and C is the unloading portion. The force difference between the

points C and D represents the pull-off force due to adhesion forces between the

elastomer particle and the glass platens. The effects of applied strains (or the

dimensionless approaches), f, and strain rates have also been examined and the

measured pull-off forces are shown in Table 4.1. By using the equation (4.21)

from the JKR theory, the work of adhesion, can be specifically calculated to be

82 + 14 mJ m 2 when the rate of pull-off is 1 jtm sec 1 . Thus the surface free


energy of the poly(urethane) elastomer may be estimated as ca 40 mJ m 2 , this

is a value which is consistent with those reported in the literature (Wu 1982).

The measured work of contact adhesion, or autoadhesion, between polymers and

smooth glasses has been reported in a number of other studies. Most of these

values are about 100 mJ m 2 (Tabor 1987) and are consistent with the current

experimental results reported in the Table 4.1. The results obtained also show

the work of adhesion is independent of the levels of the applied strains. However,

there is a detectable influence of the rate of separation (again see Table 4.1).

Many studies (Greenwood & Johnson 1981; Andrews et al. 1982; Maugis

1985; Kendall 1988; Briscoe & Panesar 1991; Kendall 1994) have shown that if

there is a significant viscoelastic contribution to the deformation of elastomers,

then the magnitude of the autoadhesion will be affected by the rate at which the

interfaces are separated. In this case, the measured work of adhesion, F, may be

more than the thermodynamic work of adhesion,Ly, often by several orders of

magnitude. Hence, the data shown in Table 1 represent not the thermodynamic

work of adhesion, but rather the measured work of adhesion. In this study, when

the rate of pull-off was increased 5 times (5 j.im sec 1 ) then the measured work of

adhesion increased about 17 %. This result shows the viscoelastic contribution

to autoadhesion. The effect is apparently small for the present system.

The "jump" into contact noted at point A is reminiscent of the phenomena

t utilised by Tabor and Winterton (1969) to measure long range Van der Waals

surface forces between micas. The jump distance (H0 ) may be simply calculated

by the equation (shown as below) proposed by Israelachvili and Tabor (1972);

nA

H0 = [6K/R](4.29)

where A is the Hamaker non-retarded force constant; n is the power law of the

tThe details of "jump contact" phenomenon are reported in Appendix A.


73A

The measured force in the loading curves between the points A' and B appears

not to smoothly increase with the imposed displacement (see Figure 4.4). This

behaviour may be attributed to the roughness at the interfaces between the sphere and

the two parallel glass surfaces. When the imposed displacement increases by a small

increment, the contact area is initially restrained by the asperity of the roughness from

moving freely to the curvature matching between the sphere and the glass. Sudden

expansion in the contact area may then occur following to a further increase in the

displacement, thereby the force increasing does not behave monotonically. The same

phenomenon may happen in the unloading process. The non-smooth decreasing of the

measured forces shown in the unloading curve between the points B and C, may also

be attributed to this mechanism.

600

z.

4000


1000

800

200

0

-200

uispiacemenr wm,

Figure 4.4: Typical loading/unloading curve for the small deformation experi-

ment. Starting (5); contacting (A); loading (AB); unloading (BC); pull off (CD);

ending (E). The members "jump" (see Appendix A) into contact at A. At A'

the applied load is zero. The pull-off force corresponds to the difference between

the magnitude of the forces at points C and D.


Table 4.1 Effects of strain (dimensionless approach) and strain-rate (pull-off speed) on adhesions of microscopic Poly-

(urethane) spheres (300 pm) in contact with two glass plates

Dimensionless approach Strain rate Pull-off force The work of adhesion(%) (pm sec_i) (pN) (mJ m_2)

4.1 + 0.1 1.0 58 + 9.8 82 + 146.8 ± 0.1 1.0 58 ± 9.8 82 + 144.1 ± 0.1 5.0 68 ± 9.8 96 + 14


law of the force (usually n=2.0+0.1 for the non-retarded cases); K is the stiffness

of cantilever beam. In current instrument the stiffness of the cantilever beam was

about 1.67x103 N m 1 , and the Hamaker non-retarded force constant is assumed

as 1.35x10' 9 J. Then, the jump distance, H0 , can be calculated as about 1.7 nm.

This result is much smaller than the experimental jump distance deduced from

Figure 4.4 which is of the order of 700 nm. Hence the negative force exhibited

around point A cannot be simply attributed to the action of a long range Van

der Waals surface force. It is notable that the noise shown in figure 2 for the

force beam, which is a manifestation of the vibration of the beam, is significantly

less than the observed jump distance. However, it is likely that additional higher

amplitude "spike" vibration noise is present which is beyond the sensitivity of

the detection system. Thus, we consider that the jump may be partly promoted

by a random high amplitude movement of the force transducer. Apart from this

vibration effect, the electrostatic attraction which occurs with some systems as

a result of charge separation across the interface may also play a role in this

"jump" contact phenomena (Derjaguin et al. 1978). These electrostatic forces

may be significantly greater in magnitude than the non-retarded Van der Waals

forces computed using equation (4.29). It may be noted here that no attempt

was made in this study to dissipate the electrostatic charging of the substrates

which were both good electrical insulators. At present the origin of this large

"jump" force, real or apparent, is unresolved '.

The interrelationships between the loads and the dimensionless approaches

'Professor K. L. Johnson (private communication) has recently analysed the "jump" phe-nomenon by considering the details of the force displacement characteristics of this particular

system; essentially by an examination of the form of Figure 4.5. His analysis, which appears to

be both correct and attractive in principle, correctly predicts the observed "jump" distances.

Essentially he includes effects due to the relaxation of the lower contact and also the trans-

ducer beam and concludes that the observed 'jumps" distance is of the order of one half ofthe distance OA. This analysis is considered further in Appendix A.


under very small deformations ( dimensionless approaches from -1.0% to 1.0%)

have been investigated in detail by both theoretical and experimental studies.

The Figure 4.5 shows the theoretical predictions for both the JKR and Hertz's

theories as well as the current experimental data obtained at different pull-off

speeds (1 and 5 im sec'). Due to the experimental uncertainty in detecting

the initial contact point between the glass platen and the poly(urethane) sphere

(see earlier) an offset correction to the experimental data is required. The data

(Case A) is the experimental data translated from the original measured data

by shifting a displacement of 0.6 1um which represents this arbitrary zero point

correction; the magnitude of the correction is of the order of the "jump" distance

mentioned previously. The results shown in Figure 4.5 clearly demonstrate that

there is a good agreement between the JKR theory (see the equations (4.24) and

(4.25)) and experimental data, especially for the behaviour under the negative

loads (tensile forces) which cannot be predicted by the Hertz theory. The points

P and Q shown in the Figure are the instability points which represent the

minimum dimensionless approach and minimum force, respectively and may be

expressed as below (Maugis 1992): Point P;

I27ir 2 y 2 (1 - y2 2 1/3

== - { 64E ]

(4.30)

Pp = —5/6irzyR

(4.31)

and Point Q;

* 137r2Ii'2(1-2 1/3

= - [_64E2](4.32)

P = P = —3/2iryR (4.33)

According the predictions of the JKR theory, the application of a tensile (pull-

off) load to the platen causes the radius of the contact area to shrink further,

and then the situation becomes unstable and the surface separate at point Q. If,


instead of controlling the load, we control the approach of c between the solids

the adhesive contact is stable down to point P (Johnson 1985; Maugis 1992)

Hence, the experimental data which closely follow the theoretical curve up to

the instability points are consistent with the predictions of the JKR theory.

Some studies (Fuller & Tabor 1975; Johnson 1975; Briggs & Briscoe 1977)

have shown that the presence of surface roughness, contrary to the viscoelastic

effect, may reduce the adhesion. According to these experimental observations,

the adhesion force between a very smooth rubber sphere and a hard counterface

prepared with various degrees of surface roughness, depends upon the counter-

face roughness and the modulus of the elastomer but not the contact geometry.

Actually, these analyses assume an infinite array of contacting asperties; see

later.

Fuller and Tabor (1975) introduced an 'adhesion parameter' 0 which is the

ratio of the adhesion force of a sphere (actually an asperity radius, /3) and the

elastic force needed to push the sphere to depth, o, into the solid, where o was

taken as the distribution of the asperity heights. An explicit expression has been

given as below:Ecr312/3"2

0 = z-y3

(4.34)

A value of 0 > 10 corresponds to the case of very small adhesion. The adhesion

force between the smooth poly(urethane) sphere and a rough brass platen was

also measured in the present study. The adhesion force in this case is less than

the force resolution of the instrument (10 tiN). If the Young's modulus and

z.-y values are assumed to be about 2.25 MPa (which is the value described

in Section 4.4.2.1) and 100 mJ m 2 , respectively, then the above relationship

may be tested. The asperity radius (/3), measured optically via the calibrated

camera system, is about 4 sum. Then, the surface roughness, o, may be estimated

to be greater than 0.9 tim. This result is consistent with an optical estimate

coo coA OQO

-0.005:)A

o

o:o0.005 0.01

0 cD..-ö0.015

zECDC)

0U-

-0.01


0.2

Hertz theory00

JKR theory- A

0 CaseA -, £0.AOO

A CaseB

0.1

0

-0.1

cx*

Figure 4.5: The comparison of the theoretical (predicted by Hertz and JKR

theories) and experimental unloading curve for small dimensionless approaches(c*) . Case A; translated experimental data (by shifting a displacement of 0.6

m) for the pull-off speed 1 jim sec 1 , Case B; experimental data for the pull-off

speed 5 jim sec1.


of the roughness of the brass platen (which is about 2 tm). With regard to

the reduction of measured adhesive force, an alternative explanation has been

suggested by Briscoe (1987). Since the radius of particle, R, is comparable with

the asperity radius (3), the number of the asperities contacting with the particle

may not be treated as many, as assumed in the Johnson and Tabor model,

but few. Then the reduction fraction of the measured adhesive force may be

expressed as /3/R; simply by adopting R as the mutual radius of curvature in

equation (4.22). Hence, in the current case the measured adhesive force may be

reduced to 3 % of the original adhesive force (assuming an asperity radius of 4

m for a point contact), which is beyond the sensitivity of the detection system.

4.4.2 Mechanical responses

4.4.2.1 Compressive compliance

The predicted theoretical curves of the loading force versus the dimensionless

approach for both the Tatara and the Hertz theories, for a 270 m diameter

poly(urethane) sphere, are shown in Figure 4.6; neither analyses considers the

contribution of provided by autoadhesive forces. Also shown in this Figure

are the experimental results obtained for a single elastomer particle 270 m

in diameter. The Poisson ratio of the poly(urethane) spheres examined in the

current study was experimentally found to be 0.48 (see Section 4.4.2.3) and this

value has been used in these theoretical calculations. The particle radii were

measured optically via the calibrated video camera system.

Figure 4.6 shows that the predictions of the conventional Hertz theory leads

to significant errors when the dimensionless approach exceeds about 10 % in

comparison with both the experimental data as well as the predictions of the

Tatara theory. Since c is equal to a2 /R2 (a is the radius of the contact area) , the

above result implies that in the current case the Hertz theory is accurate for a/R

Adhesion Force -Hertz load -


values up to 0.33. Johnson (1985) has shown that for experiments on "Araldite"

(a commercial epoxy resin, Ciba Geigy, UK) spheres the Hertz approximation

was adequate upto values of a/R 0.3. For larger approaches, the experimental

data and the predictions of the Tatara analysis are in reasonable agreement.

By using the Hertz theory, a Young's modulus of 2.25 MPa has been cal-

culated based on a linear least square fit of the imposed loads as a function

the 3/2-power of the approach using the experimental data in the range 5% up

to 10% deformation (dimensionless approaches). However, by using Tatara's

theory with a minimum least-squares (MLS) fitting for a much wider range of

experimental data (upto 37 % deformation) the corresponding initial Young's

modulus, E0 , was found to be 2.06 MPa; a difference of circa 10%. This slight

difference is primarily due to the over estimation of the strain dependency of the

Young's modulus introduced by the Tatara theory, at large deformations. Thus,

the Tatara theory with a MLS fitting, may be regarded as a successful method

for the characterising of the Young's modulus of these micro-elastomeric spheres

for a large range of deformations.

Since the Hertz theory and the Tatara analysis both neglect the adhesion

force between the elastic bodies, it is appropriate to estimate the influence of

this force upon the evaluation of the magnitude of the compressive compliance

for the large deformations introduced into the present system. An explicit re-

lationship, derived from the JKR theory by Johnson (Johnson 1993), suggests

that the ratio of adhesion force to the Hertz load is as follows;

PaPaP1—P1(2h/2 - (6RLy'\"'2 -

2\¼P1 ) P1 1 - (4.35)

where Pa ( =/67rAyRP1 ) is the effective force of adhesion. The three dime-


0.05

0.04

0.03

0.020

0.01

0.00

—0.010.0

Hertz theory

- Tatara theory

Tatara theory with MLS fitting

o Experimental data

0.1 0.2 0.3 0.4*

a

Figure 4.6: The comparison of experimental and theoretical load versus di-mensionless approach, &, curves for a 270 urn poly(urethane) sphere. The

experimental data and various predictions are shown. The Young's moduli are

respectively 2.25 MPa for the Hertz predictions and 2.06 MPa for the Tatara

analysis. The Poisson's ratio is taken as 0.48.


nionless variables, F, a and P1 , are defined as below;

P = (37rLR)

(4.36)

aa = (4.37)

(9YR21_u2))h/34E

= a (4.38)

From Figure 4.5, the value of P is about 50 N. Hence, the equation (4.35)

gives an estimate that the ratio of the adhesion force to the Hertz load as about

10%, when P1 is approximately 0.02 N where the corresponding dimensionless

approach (about 25%) is far beyond the satisfactory fit of the Hertz region in

Figure 4.6. Hence, in the Hertz region when neglecting the autoadhesive forces,

we introduce significant (at least more than 10%) errors in the evaluation of

the compliance. However, at the larger deformations (30%), the value of the

compressive force is about 24 mN as calculated by Tatara theory and the value

of the adhesion force is about 2.3 mN computed by using the equation (4.23).

Since the contact area for these large deformations may be over estimated by the

JKR theory, the true adhesion should be much less than 2.3 mN. It thus seems

that the neglect of the adhesion force contribution in the Tatara theory may

cause the appreciable errors to some, but an unknown, extent, in the evaluation

of the elastomer microsphere compliance.

It is physically intuitive that, at very low applied loads (P << 3/2iryR),

the magnitude of the adhesion force, as compared with that of the mechani-

cal force proposed by Hertz, has a greater contribution to the deformation of

microspheres than large spheres; the compressive force is proportional to the

square of the radius at a constant dimensionless approach (see equation (4.39)

and (4.40)), while the adhesion force is directly proportional to the radius, (see

equation (4.23)). Equation 4.35 express this effect in a quantitative manner.


4.4.2.2 Scaling effects

The scaling effects for the Herzian contact have also been investigated by com-

paring the theoretical and experimental dimensionless applied load versus dimen-

sionless approach curves for a 38.3 mm poly(urethane) sphere and the micro-

poly(urethane) spheres (300 jim) (Figure 4.7); a variation in the radius of a

factor of approximately one hundred. The theoretical curve is predicted from

the Hertz theory which shows that the reaction force is proportional to the

3/2-power of the dimensionless approach. Hence, the interrelationship between

the dimensionless approach and the dimensionless applied force, P/K', may be

expressed as;

*3/2

where K'is,4ER2

3(1 - u2)

(4.39)

(4.40)

However, the dimensionless applied forces of these two spheres are both under

estimated by the Hertz theory for the larger dimensionless approaches. The

above results show that this description of the scaling effects may be reasonably

valid in the medium range of the deformation regime for the Hertzian contact

region. However, since the formulation of the Tatara theory is intrinsically

complex, there is no explicit solution available to describe the interrelationship

between the dimensionless applied load and the dimensionless approach for the

larger deformations.

4.4.2.3 Geometric features of deformations

The dimensionless central lateral extension (the ratio of U(R) to R), U*(R),

versus the dimensionless approach curves shown in Figure 4.8 were obtained


0.08

0.06

0.04

0.02

0.00

v Micro PoIy(urethane) Sphere (300 /tm) v• Large Poly(urethane) Sphere (38.3 mm)

VTheoretical Prediction (by Hertz Theory) vv

VV.VV

VV VSV

VVVV •

VVV

V •

VV\1i7••

- •

I I I I I I

—0.01 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07

* 3/2a

Figure 4.7: Comparison of the theoretical and the experimental dimensionless

applied load, P/K, versus dimensionless approach (at) curves for 300 jm and

38.3 mm poly(urethane) spheres.


both experimentally and also computed using the Tatara theory. The dimen-

sionless central lateral extensions of the 240 im and the 38.3 mm poly(urethane)

spheres are both under-predicted in the small deformation regime by the Tatara

theory. Some of the original photographs of the deformed micro-sphere which

were visualised from the bottom view are shown in the Figure 4.9. This dif -

ference may arise from the constant volume assumption adopted by the Tatara

theory; essentially the Tatara analysis assumes a constant Poisson ratio. For

large deformations good agreement was observed.

Using Tatara's theory, a family of lines for the computed dimensionless ap-

proach versus dimensionless central lateral extension for different chosen Poisson

ratios values are shown in Figure 4.10. Apparently, the equations (4.12) and

(4.14) for the calculation of c and U(R) are both associated with the Young's

modulus and Poisson's ratios. However, combining equations (4.12) and (4.14)

gives an alternative relationship which only includes the Poisson's ratios and

may be expressed as below;

3Rlr(1_112)

(1 + Ba2 f(a')R (i Ba2

___ - ____ 4a' (4.41)i Ba2U (R) - U (R) = (1 + ii) (1 i- -) ( - (1 - 2v) (1 -

From these predictions, and by the use of the equation (4.41), the Poisson ratios

of a single elastomeric spheres may be simply characterised by measuring the

two major experimental deformation parameters; the dimensionless approach

and the dimensionless central lateral extension variation. An explicit numerical

correlation between the Poisson ratio, ii, and the dimensionless central lateral

extension, U5 (R), for the deformation at 40% dimensionless approach are pro-

vided in the following equations

Li (US (R)) = 81.163 (U* (R)) 3 - 38.535 (U* (R)) 2 + 6.989 (U* (R)) + 0.04242

-

(0.0485 U5 (R)) (4.42)


0.4

• Micro PoIy(urethane) Sphere (240 i.Lm)

V Large Poly(urethane) Sphere (38.3 mm)- Theoretical Prediction (by Tatara Theory)

0.3

0.2

0.1

0.0'0.0 0.1 0.2

0.3 0.4 0.5 0.6*

a

Figure 4.8: Comparison of the theoretical and the experimental dimensionlesscentral lateral extension, U*(R), versus dimensionless approach, &, curves for240 ,um and 38.3 mm poly(urethane) spheres.

(a)

(c)

(b)

(d)


Figure 4.9: Photographs of the deformed elastomeric sphere (ca 240 j.im) from

a bottom view for various dimensionless approaches (a); (a) a=0 % (b) a 9.5

% (c) a= 28.3 % (d) a= 55.8%.


v (U* (R)) = —126270 (U (R)) 4 + 12173 (U (R)) 3 - 473.2 (U* (R))2

+ 16.94 (U* (R)) - 0.0984; (U*(R) < 0.0485) (4.43)

Thus, for the poly(urethane) materials which have been used in the current

study, the value of Poisson's ratio was computed as about 0.48 + 0.007 based

upon the experimental data shown in the Figure 4.8. Then, another important

mechanical property, the bulk modulus K5 ( = E/3(1-2v) ), may be calculated as

about 20 MPa. This value seems to be too small for these elastomeric materials

based upon accepted norms. The bulk modulus of most elastomeric materials is

usually about 1 CPa; to achieve such a high value, the value of Poisson's ratio

should be at least higher than 0.4997. However, it should be pointed out that the

uncertainty in Poisson's ratio value determined by the current method (shown

above as 0.007) is a nominal uncertainty which does not include the potential

error sources contributed by the limited resolution of the optical system, the

numerical truncation errors and the quasi-spherical shape of the elastomeric

spheres examined. The latter effect is a significant factor, but is not a source

of error which is readily quantified. Thus we may conclude that although the

Poisson's Ratio may be computed by these means the potential uncertainty is

high for these incompressible materials.

The contact area measurement has been the key experimental variable used

in the evaluation of both the Hertz's and the autoadhesion theories; the initial

motivation for the development of the JKR theory arose from a desire to explain

contact area data. Though the measurement of the area of contact has been

proven to be viable in large scale macroscopic experiments, it has proven to

be far more difficult at the microscopic scale. Due to the diffraction limit,

the resolution of conventional optical systems is, at best, 0.25 ,um. However,

practically it can often be difficult to resolve dimensions of less than 1 jim and

4.4 Results and discussions 89A

The error in the estimation of the Poisson ratio for the sphere may be raised from

the inappropriate boundary condition adopted by the Tatara theory, such as the

frictionless interface between the sphere and the plates. The influence of the friction

force, as mentioned in detail in section 4.4.1, which restrains the contact area from being

freely conformed to the plates, may cause variation in the central lateral extension

measurements. In order to examine quantitatively this effect, a further study may be

suggested to be carried out by compressing the sphere with well-lubricated platens, such

that the effect of asperities may be eliminated.


1.2 I I Iv=0.5

1.0 -

0 . 8 -

0.6 -

0.4 -

0 . 2 -

0.0 -

—0.2 I I I0.0 0.2 0.4 0.6

*a:

Figure 4.10: Simulated dimensionless central lateral extension, U*(R), as a func-

tion of dimensionless approach, c4, for various Poisson ratios (v) computed by

the modified Tatara analysis.

3

=0.1

v=-0.3v=-0.5

v=-0.8

0.8 1.0 1.2

4.5 Summary 91

other workers have also noted that the precise measurement of the contact area

for microscopic systems has proven to be intractable (Evans & Skalak 1979). For

a 100 zm sphere the contact diameter is calculated as ca 4 um for zero applied

load. Hence, the resolution of optical system would result in an error of possibly

as large as about 25 %. We may thus conclude that contact area measurements

are generally not viable as a means for accurately characterising these systems.

4.5 Summary

For the present micro elastomeric spheres, at small deformations, the influence

of autoadhesion, as modelled here by the JKR theory, predominates the contact

compliance. Theory and experiment are in good accord. Whilst in the case of

large deformations for non-linear elastic materials the lateral extension of the

compressed sphere, as modelled by Tatara theory, plays an important role in

governing the gross deformation. Again, theory provides a good description of

the experimental data. The lack of a consideration of autoadhesion in both the

Hertz theory and the Tatara theory may cause significant errors in the eval-

uation of the compliance in the small imposed deformation region. The JKR

based analyses, combined with the experimental data of the measurements of

the pull-off forces, allows the apparent interfacial free energy to be accurately

determined. Furthermore, the results have also shown that the presence of sur-

face roughness may reduce the apparent adhesion; these viscoelastic materials

exhibit an increased apparent interfacial free energy with the rate of separa-

tion. The Tatara based analysis, combined with experimental data, at a 40%

dimensionless approach, allows the explicit estimation of the Poisson Ratio for

these elastomeric particles. The potential errors are however significant. The

application of the Hertz or Tatara analyses, depending upon the dimensionless

4.5 Summary 92

approach regime experimentally investigated, also allows the Young's modulus

for a single particle to be accurately determined.

Chapter 5

The deformation of liquid-filledmicro-cellular entities

This Chapter describes an experimental and theoretical study of the compressive

behaviour of single microcapsules; that is liquid filled cellular entities (ca 65 ,um

in diameter) with a thin polymer membrane wall, upto a dimensionless approach

of 60 %. The major geometric deformational parameters such as central lateral

extension as well as the failure phenomena are reported and recorded through

the microscope based visualisation system. The elastic modulus, the bursting

strength of the membrane and the pressure difference across the membrane are

computed. All these parameters are critical physical properties for both the

manufacture and usage of microcapsules. Using the current methodology these

physical properties can be determined by this experimental approach combined

with a theoretical analysis which is also presented in this Chapter. This theo-

retical model which was originally developed by Feng and Yang (1973) considers

the deformation of a nonlinear elastic spherical membrane which is filled with

an incompressible fluid. Furthermore, this approach may be considered as the

foundation for extending such methods to more complicated quasi-spherical nat-

ural entities, such as osmotic-swollen red blood cells or plant cells (see Chapter

93

5.1 Introduction 94

6.).

5.1 Introduction

Microcapsules, that is liquid filled cells with thin membrane walls, are beginning

to be used in the pharmaceutical, agriculture, food industries, as well as in

biomedical technology industries for a wide range of applications from drug

delivery to the construction of synthetic cells for artificial organs and artificial

blood. The mechanical properties of these synthetic microcapsules have been

recognised to be important not only for determining the kinetics of the release of

encapsulated chemicals but also for controlling the durability of the products in

processing and use. The process of compacting microcapsules via compression

is considered as a convenient method for assembling microcapsule particles into

a dosage form. However, the individual particles must exhibit sufficient physical

integrity in order to withstand processing whilst maintaining the required drug

release profile in their final dosage form (Loftsson & Kristmundsóttir 1993).

Moreover, some studies (Jalsenjak ct al. 1976; Prapaitrakul & Whitworth 1990)

have shown the pressure difference across the wall membrane may be one of

critical factors which effects the rate of drug release from tableted microcapsules.

Of course, liquid-filled microcapsules exist ubiquitously in biological systems in

many and various forms. Some authors (Halpern & Secomb 1992) consider

the mammalian red blood cell motion in the microcirculation environment to

be analogous to the squeezing of liquid-filled membrane systems between two

parallel plates.

There are several experimental methods which have been reported for char-

acterising the deformation behaviour of microcapsules. Jay and Edwards (Jay &

Edwards, 1968) have measured the elastic properties of the membrane of micro-

5.1 Introduction 95

capsules by using the micropippette aspiration technique. (Chang & Olbricht

1993) have studied the elastic properties of the membrane by observing the mo-

tion and deformation of a synthetic, liquid-filled capsule (diameter about 2 to 4

mm) that was freely suspended in a hyperbolic extensional flow. However, this

method has not been applied to microcapsules. Recently, (Zhang et al. 1994)

have measured the bursting strength of microcapsules by using a micromanip-

ulation technique which squeezes a single microcapsule between two platens.

Since the position of the moving platen, in their work, cannot be accurately

determined, a detailed study of the deformation behaviour of the microcapsules

is currently difficult using this method.

Up to now, there have been no suitable experimental techniques and associ-

ated theoretical models which permit the simultaneous study of the elasticity, the

tension distribution and bursting strength of the microcapsule membrane and

the pressure difference across the membrane. However, to achieve these goals,

the micro-upsetting method may be considered as a powerful tool to accurately

measure the force-displacement curve and the major geometric deformational

parameters. Through the quantitative analysis of the interrelationships between

the force and the displacement, as well as the prediction of the geometric defor-

mational parameters the above critical physical information may be estimated.

Several investigators have theoretically modelled the contact mechanics of

spherical nonlinear membranes. Feng and Yang (1973) were first to consider

the problem of the deformations and the stresses in an inflated nonlinear elastic

spherical membrane compressed between two frictionless rigid plates. Then,

Larden and Pujara (1980) extended the analysis to a membrane filled with an

incompressible fluid and were able to accurately predict the deformation of the

sea-urchin egg, compared with the experimental results which were previously

reported by Yoneda (1973). Later, Taber (Taber 1982) carried out experimental

5.2 Experimental 96

work and proposed a similar theoretical model which included the wall bending

moment in the governing mechanisms of the compression of fluid-filled spherical

shells by rigid indenters.

5.2 Experimental

5.2.1 Material

The microcapsules which have been used in this current study were a water/oil

multiple emulsion drop contained within a thin polymeric membrane (see Figure

5.1). The membrane wall is, in this case, made of a poly(urethane) elastomert.

The diameter of the particles varied between 50 and 100 ,am and the wall thick-

ness, estimated by scanning electron microscopy following freeze fracture, ranged

between 1 and 2 jim. These materials were provided by Zeneca Ltd, UK.

5.2.2 Experimental procedure

A petri dish was placed underneath the microscope lens. A liquid film, isotonic

to the liquid contained within the microcapsules, was then placed in the dish.

Microcapsules were randomly drawn into a micropipette and then carefully dis-

charged into the petri dish. A particular microcapsule was chosen at random

for the test. In order to confirm that only one particle was in contact with the

microplaten, the contact region was firstly examined using a low magnification

lens on the microscope. The microplaten was then slowly driven up and down to

find the initial contact point between microplaten and the selected microcapsule.

When the microplaten and the microcapsule initially touched, the microcapsule

was slightly shaken and this effect was monitored in the video image of mi-

crocapsule shown on the video monitor. The microplaten was then driven at

tThe preparation method for these microcapsules is reported in detail by Brown (1993).

5.2 Experimental

97

Polymer encapsulant

\ \ Water (W)

OIIOQQ )

Figure 5.1: Schematic representation of the microcapsules.

5.3 The theoretical analysis 98

a constant speed, about 2 im per second, during the microcapsule compres-

sion. The imposed force and the displacement of squeezed microcapsule during

loading and unloading were automatically recorded through the data acquisition

system linked with the personal computer. Also, a video image of the deformed

shape of the microcapsule was video recorded so that the lateral extension could

be determined after the experiment. The maximum imposed displacement was

gradually increased until bursting of the microcapsule was observed.

5.3 The theoretical analysis

It is well known, in the elastic deformation theory of plates and shells, that the

bending rigidity (D), or the flexural rigidity, of a thin isotropic plate or a thin

shell is proportional to the cubic power of the wall thickness. Thus, the bending

rigidity may be expressed as:

DEh3

- (1_v2)

where E is Young's modulus, h, is the wall thickness, and ii is Poisson's ratio.

In contrast, the extensional rigidity is Eh, and proportional to the first power of

the wall thickness. When the wall thickness, h, is very small, the contribution

to the sensed rigidity from the bending rigidity is much smaller than the contri-

bution from the extensional rigidity (Fung 1993). Furthermore, Taber's (1982)

experiments and calculations for a thick shell (the ratio of radius of thickness

is 6.0) demonstrated that when a point load was applied the bending stress

governs the behaviour at small deformation (the dimensionless approach is less

than 20 %), but that the membrane extensional stresses dominate at larger de-

formations. Since the microcapsules which were used in this study have a high

ratio of radius to thickness (about 16); i.e. thin wall systems, the response forces

(5.1)

(5.3)

(5.4)

aw* IA12C1 A 1 A 2 -

9A 1 =aw*

2C1A1A2 (;- -ôA 2 =

i(i +

1 2/i\

AA) (1+ )

=

U2 =

(5.5)

(5.6)


contributed from the bending moment may be reasonably neglected in the first

instance. This assumption will be further discussed in section 5.4.1.

The constitutive equations (relationships between stress and strain) used in

this study to represent the behaviour of the microcapsules membranes are those

for Mooney-Rivlin and Neo-Hookean materials which have rubberlike non-linear

elasticity (Mooney 1940; Rivlin 1948).

In the Mooney-Rivlin model the strain-energy function*, W, of an isotropic

incompressible material, is

W = C1(11-3)+C2(12-3)

= C1[(11-3)+3(I2-3)]

(5.2)

where C1 and C2 are the material constants with the dimensions of stress, fi =

C2/Ci ; for a homogeneous and isotropic, incompressible elastic material, C1 is

equal to 6E. I and 12 are strain invariants which may be expressed in terms of

the principal stretch ratios, ) and X 2 , and are shown as below:

1'1 =

)1 1

j -2 - 1 2++

Then the principal stresses in the meridional and circumferential directions, a1

and a2 (see below; Figure 5.2), respectively, may be expressed as

For a finite deformation, the relationship between the stress and the strain of a rubberlike

material is non-linear and a more appropriate formulation of the constitutive equation is

to consider the strain energy function, W* , of the material. The strain energy function is

identified with the internal energy per unit volume in an isoentropic process, or the freeenergy per unit volume in an isothermal process.


For the configuration of this spherical membrane system, the relationship

between the stress resultants, per unit length, in the meridional and circum-

ferential directions of the deformed surface, T1 and T2 , respectively, and the

principal stresses o, a2 , may be expressed as below (Feng 1973):

hcr1 1 "T1 = = 2hC1

-( i + (5.7)

A 1 )'2

ha2 "A2 1 "T2 = = 2hC1

A 1 A 2 - AA) (i + 1i3A) (5.8)

where h is the initial thickness of the membrane.

A Neo-Hookean material description may be seen as a simplification of the

Mooney-Rivlin formula by assuming 3 is zero (Rivlin 1948). For a Mooney-

Rivlin material the value of /3 has been taken as 0.1, as suggested by Green and

Adkins (1970), in the current study.

Lardner and Pujara (1980) derived two groups of governing equations for

two separate deformation regions: the plate-membrane contact region and the

non-contact deformation region. The details of the derivation of these equations

are given elsewhere (Feng & Yang 1973; Lardner & Pujara 1980). The final

results are summarised below:

'The equation (20) in Feng and Yang (1973) contains an error and may be corrected asbelow:

1 A1 A 2 cost,b A 1 3A 1 =

(+)(1+A ( inb[(j (1+aA)

A 1 1 1 A1 1(A 2A1 \ 1 1 1—2A2 -- 3)J + A 2 sirn,b R.- -) (.AA2 A1A)J)

(5.9)(A2 1 2/

The equation (2.2) in Lardner and Pujara (1980) should be corrected to be the same as thatin Equation 5.12 of this Chapter. Also the equation (2.10) should be corrected to read as

A 1 1 \

(A2 1 2/= 2hC1 - - -----) ( 1 + 1'A)

A 21 \

(A, 1 2/

T2 = 2hC1 - - ) ( 1 + tA) (5.10)

The definitions of and t reported in their cited papers are the same as that for the parameter3 in the current study.


101

Contact region

A 1 1f3'\ fA1—A2cos\ 1'f2\A2sin ]) - sinb ) -)

(5.11)

A1 - A2cosA2 =

(5.12)sinb

and, Non-contact region

(5cos - Wsin\ ff2) (W\A1

= Sifl2,L) ) - '-) ()(5.13)

8 I =W

(5.14)

1/2 D

WI = AW + (A 1 - W2) /T2 \ A 1 (A - W2 ) r0

(5.15)A 1 ()- T1

where,

- ôT1 - 2hC1 (i + x) (- +

)(5.16)-

=- 2hC1 [(3,\4 - 2) (1 + + 2i9A2 A1

3 ].i'

IA 1 A2/1 1= T1 - = 2hC1- — 3

- AA)](5.18)

and where P is the pressure inside the membrane after initial contact; the primes

indicate differentiation with respect to , the angular position reference in the

undeformed sphere. A schematic diagram of the half spherical membrane, be-

fore and after contact is shown in the Figure 5.2. The Figure also shows the

spherical coordinates (r,e,) used for the spherical membrane before contact

and cylindrical coordinates (p,e,i7) which are used for the description of the

deformed membrane after contact. The variables S and W are defined as:

102

(5.19)

(5.20)

(5.21)

(5.22)

(5.23)

(5.24)

(5.25)


8 = A2sin

W=6'

The boundary conditions for this problem are;

=

= F; = F (i)noncontact(6

= F; (A2)contactF SjflF)

=

=

where F is the contact angle; figure 5.2.

Since the original boundary-value problem has been transformed into an mi-

tial value problem, the governing equations (equation (5.11) to (5.15)), with

their boundary conditions, can be solved by a standard numerical scheme; the

Runge-Kutta method (Ferziger 1981). Extending the works of Feng & Yang

(1973) and Lardner & Pujara (1980), an independent computer algorithm was

developed as described below (Figure 5.3). The calculation procedures of Feng

& Young and Lardner & Pujara, both predescribed the angle of the contact

area (F) and then calculated the other parameters which include the distance

between the rigid plate and the equator of the spherical membrane after contact,

ij. However, the angle of contact area has been proven to be difficult to mea-

sure by experimental observation in these microscopic systems (Yoneda 1973).

Hence, it is appropriate to modify their procedures so as to prescribe ij and then

to calculate the other parameters. The assumption, suggested by Lardner & Pu-

jara, that is the volume of encapsulated solution is constant has been adopted

p View

Sphere after deformation Q(ro,O, 1/,)

Q '(p,E,ij)

p

ound


I I Side ViewSphere before deformation ,_ i

Figure 5.2: Geometry for the contact problem for the half of a thin wall spherical

membrane between two large rigid plates.

5.4 Results and Discussions 104

in the current analysis. The flow chart of the enhanced algorithm, applied in

the current study, is presented in Figure 5.3.

5.4 Results and Discussions

The above theoretical analysis, combined with the corresponding experimental

data, has provided a route for determining the elastic modulus of the membrane

(section 5.4.1), the internal pressure (section 5.4.2), the tension distribution on

the membrane (section 5.4.3), and the geometric features of the deformation

(section 5.4.4). By a comparison with the experimental observations, the basic

assumptions adopted in the theory, outlined above, may be tested.

5.4.1 Elastic modulus of the membrane

The loading/unloading curve for a 65 m microcapsule up to a 60 % deformation

(dimensionless approach) has been investigated and is shown in Figure 5.4. The

dimensionless approach parameter, y, comprises the distance between the rigid

plate and the equator of the spherical membrane after contact, i, the stretch

ratio of the initial inflation, ), and the radius of the undeformed microcapsule,

r0 and is of the form: (1 - i7/roA 3 ). Since in our case the liquid in the petri dish

is isotonic with the liquid contained in the microcapsules, it seems reasonable to

assume that the initial inflation is zero; i. e. that \, = 1 although this condition

is unproven. Based upon optical observations during the loading process, the

microcapsule started to burst when the deformation reached about a value of 58

% deformation (see Figure 5.5(d)). The unloading curve, after the burst, shows

that the reaction force is now very small compared with the correspond loading

value, especially at large deformations. This observation implies that the reflex

force contributed by the bending moment, after rupture, is insignificant during

the post rupture deformations and this is consistent with the earlier assump-


Start

Prescribe j

Assume contactangie r

Assume X1)

Solve the governingequations of contact regionby the Runge-Kutta method

Assume P

Solve the governing equationsof non-contact region by theRunge-Kutta method

-check 3=O at V7/

Check constantvolume

YES

Stop

Figure 5.3: The flow chart for the enhanced version of the algorithm for solving

the set of the equations of the membrane model used in the current study.


tion of a mainly tensile membrane response; that is the bending contribution is

negligible.

Experimental loading/unloading transverse compliance curves are shown in

Figures 5.6 and 5.7. In these Figures the force, F, has been non-dimensionlised

to be in the dimensionless form: y = F/C1 hro). The theoretical predications

based upon the membrane model with Neo-Hookean and Mooney-Rivlin consti-

tutive equations, combined with the minimum least-squares (MLS) fitting to the

experimental data, are also shown in the Figures 5.6 and 5.7. The extensional

rigidity of the membrane, Eh, may be calculated to be 538 N m 1 for a Neo-

Hookean material and 536 N m 1 for a Mooney-Rivlin material, respectively,

from the MLS fitting between the theoretical predictions and experimental data.

The difference is small. Hence, if the thickness of the elastomeric membrane wall

is assumed to be 2 ,um, then the elastic modulus of membrane can be calculated

to be 2.69 MPa for a Neo-Hookean material and 2.68 MPa for a Mooney-Rivlin

material, respectively. This result, which shows no significant difference be-

tween the estimated Young's modulus obtained by the Mooney-Rivlin and the

Neo-Hookean laws, suggests that the second term in the Mooney-Rivlin law,

/3(12 - 3), may not important for the description of the deformation of these

elastomeric membranes. We may also noted that a value of the elastic modulus

2.69 MPa is a sensible one for this type of polymeric membrane.

Compared with the experimental data, the theoretical calculation of the

loading/unloading curves obtained from the membrane model with either Neo-

Hookean material law or Mooney-Rivlin model are slightly under-predicted when

the deformation is below 15 %. This difference may arise from the fact that the

bending moment still has influence, to some extent, when the deformation is

small (Taber 1982). For the intact microcapsules the force difference between

loading and unloading (shown in Figure 5.6 and 5.7), which is very small, is con-

0.6

z

q)0

0


1.0

0.8

0.2

0.0

—0.2—0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

(1 —/r0X)

Figure 5.4: The experimental loading and unloading (after bursting) curves of a

65 im microcapsule. The bursting point is near 58 % dimensionless approach,(1—/ro)t).

108

(c)


1

(a)

(b) (d)

Figure 5.5: Photographs of the bottom view of a deformed microcapsule for

various dimensionless approaches (y); (a) y=O % (b) y= 20 % (c) y= 40 % (d)

y=58%.


sistent with the assumption that the transport of encapsulated solution across

the membrane seems to be insignificant during the chosen time scale of the

loading/unloading process. Furthermore, it appears that the viscoelastic effects

associated with the deformation of the membrane, again within the chosen time

scale, are also negligible. The agreement between theory and experiment , as well

as the reasonable prediction of the membrane elasticity, would suggest that the

analyses based on both Mooney-Rivlin and Neo-llookean constitutive equations

can be generally applied to the compressive deformation of the polymer-bounded

microcapsules. Moreover, the close agreement between the theoretical predic-

tions and the experimental loading! unloading curves, as well as the reasonable

estimation of the membrane elasticity, would suggest that the analysis based on

the membrane model , associated with either the Mooney-Rivlin law or the Neo-

Hookean constitutive equations, can be applied to the compressive deformation

of polymeric membranes.

5.4.2 Internal pressure

The computed internal pressure versus deformation curve, which is shown in

Figure 5.8, indicates a weakly nonlinear increase of the internal pressure with

the imposed deformation. This predicted result is important not only for the

preservation of the capsule integrity but also for the chemical release behaviour

of microcapsules. The computed bursting pressure (the pressure at 58 % di-

mensionless approach) is about 26 KPa. This infers a rupture at a nominal

membrane strain of ca 10 % in the radial direction. This rupture strain is rather

small for an elastomeric material; a value one order of magnitude greater may

have been anticipated. The nature of the preparation of these membranes, a

chemical reaction at a liquid! liquid interface, will naturally produce many de-

fects in thickness and composition. Thus whilst it seems that the membrane is


3.0

2.5

2.0

1 .5

1 .0

0.5

0.0

—0.5—0.1

• Experimental data- Theoretical prediction

S

(Mooney—Rivlin) •

S

- .•.

•

• •S•• S •S

S...

0.0 0.1 0.2 0.3 0.4 0.5

(1 —/r0X)

Figure 5.6: The dimensionless experimental loading/unloading curve (deforma-

tion up to 40 %) and theoretical predictions produced by the membrane modelwith a Mooney-Rivlin material law. The parameter y = (F/Ci hr0A) is thedimensionless force and the quantity (1 - /r0 )) is the dimensionless approach.A is 1.0 and C1 is 16.08 MPa.

'V../.

.,fr/.

•

./ ••/.

..:

••

0.0 0.1 0.2 0.3 0.4 0.5

(1 —/r0X5)


3.0

2.5

2.0

1 .5

1 .0

0.5

0.0

—0.5—0.1

• Experimental data

Theoretical prediction •

(Neo—Hookean) .

•

Figure 5.7: The dimensionless experimental loading/unloading curve (deforma-

tion up to 40 %) and theoretical predictions produced by the membrane model

with Neo-Hookean material law. The parameter y = (F/Ci hr0 ) is the dimen-sionless force and the quantity (1 - f/r0 ),) is the dimensionless approach. A 3 is1.0 and C1 is 16.14 MPa.


not porous it may have many thin regions or effective notches which reduce its

toughness.

For the modelling of the release rate of encapsulated solutions across the

membrane under the deformation, the Darcy's law (Darcy, 1856) which was de-

rived from the assumption of laminar flow through a cylindrical channel may

be applied to the current case. Basically, this law supposes that the flux of

the liquid across a membrane is directly proportional to the pressure difference

across the wall; if the membrane is assumed to be a porous media and that

the permeability of the membrane is a constant. Hence, the release rate of the

solution for the compressive microcapsules will nonlinearly increase with the de-

formation due to nonlinear increase of the internal pressure. The microcapsules

used in some applications, such as the time-release drug delivery and for the

immobilisation of enzymes, are often required to maintain constant release rates

of the encapsulated solutions. Therefore, the above result are interesting and of

crucial value in the control of the performance of such deformed microcapsules.

5.4.3 Membrane tension

The membrane tension (the extensional force per unit wall thickness) on the

membrane can be predicted by the theoretical model. If the elastic modulus

oL,membrane is 2.69 MPa, the calculated tension profile for the case of a 58 %'thdeformation (at rupture) is shown in the Figure 5.9; also see Figure 5.2. The

result shows that the tension in the membranes is not uniform but increases

with the angular position parameter (i5). On this basis, the bursting point will

be always located on the equator; the stresses are imposed at the poles. This

predicted feature is consistent with the experimental observations noted through

the image system (see Figure 5.5(d)). This prediction also shows that, for the

present case, the resultant stress in the deformed circumferential direction, T2,


113

30

• Neo—Hookean

25 V Mooney—Rivlin

20

a.

QL0.0

0.1 0.2 0.3 0.4 0.5 0.6 0.7

(1 —/rA)

Figure 5.8: Internal pressure versus deformation curve for the compression of a

compressive microcapsule. The Young's moduli of membrane are respectively

2.69 MPa for a Neo-Hookean material and 2.68 MPa for a Mooney-Rivlin re-

sponse.


is always larger than the stress resultant in the deformed meridional, T1 . The

bursting strength, the tension on the equator for 58 % deformation, for the

microcapsule is computed as about 1.04 N cm 1 which is the stress resultant in

the deformed circumferential direction at rupture.

5.4.4 Geometric features of the deformation

The simulated deformed external shapes of the microcapsule are show in Figure

5.10. These results show that, for large imposed deformations, the noncontact

region must be significantly stretched in order to maintain a constant enclosed

volume; this condition is assumed to occur in the present case. The predicted

behaviour has been identified by measuring the dimensionless central extension

(central lateral extension/ initial radius) of the free surface from the microscope

images (shown in Figure 5.5). The comparison between the values obtained from

the experimental measurements and the theoretical predictions are shown in

Table 5.1. The results show that the theoretical predictions and the experimental

measurement are in good accord.

An interesting comparison of the computed geometric features of the defor-

mation between the homogeneous elastomeric sphere with a Poisson ratio of 0.5

(fully incompressible) and the liquid-filled spherical entities with an elastomeric

membrane, is shown in the Figure 5.11. For the dimensionless central lateral

extension, there is no significant distinction (less than 0.5 %) between these

two entities upto a 50 % dimensionless approach (see Figure 5.11(a)). However,

when dimensionless approach achieves 60 %, there is a 8 % difference between

these two entities. For a 100 um spherical cellular entity, the difference in the

central lateral extension between these two cases is calculated as 8 ,im, which

may be resolved by the current instrument. However, practically such a large

deformation may cause the liquid-filled spherical entities to fracture.

90060°


0.12

0.10

0.08

• 0.06(1,

ci,F-

0.04

0.02

0.0000

- Neo—Hookean

Mooney—Rivlin

300

Figure 5.9: Variation of the wall tension, the extensional force per unit wall

thickness, with angular position, , for a 58 % deformation. The Young's moduli

of membrane are respectively 2.69 MPa for a Neo-Hookean material and 2.68

MPa for a Mooney-Rivlin system. T1 and T2 are the stress resultants in the

meridional and circumferential directions, respectively.


1 .4

1 .2

1 .0

0.80

0.6

0.4

0.2

0.00.0 0.2 0.4 0.6 0.8

1.0 1.2 1.4

p /r

Figure 5.10: The simulated deformed shapes of a microcapsule , = 1, 3 = 0.1)

for 20 %, 40 % and 60 % deformation as used in the current study.


Table 5.1 The comparison of the experimental and theoretical dimensionlesscentral laternal extension versus dimensionless approaches

Dimensionless approach Dimensionless central laternal extension(%) Theoretical predictions / Experimental measurements

20 1.045 / 1.056 + 0.00840 1.150 / 1.152 + 0.010

AA

AA

A •A

AA

A •

A.

0.4 0.5 0.6


0.45

0.4

0.35

0.3

0.25

::: 0.2

0.15

0.1

0.05

0

0

A Homogeneous sphere(Tatara theory)

• Liquid-filled sphericalmembrane (Membranemodel)

•A

A

0.1 0.2

0.3

(a)

A

A

A

A

A

A

A

A

A.

A

0.4 0.5 0.6

A.

1.4A Homogeneous sphere

(Tatara theory)1.2

• Liquid-filled spherical

1 membrane (Membranemodel)

0.8*

0.6 A.

A

0.4 .A

0.2

00 0.1 0.2 0.3

(b)

A.

Figure 5.11: A comparison of the computed geometric features of the defor-

mation of a homogeneous elastomeric sphere with a Poisson ratio of 0.5 and a

liquid-filled spherical entity with an elastomeric membrane; (a) the dimensionless

central lateral extension versus the dimensionless approach (b) the dimensionless

contact radius versus the dimensionless approach

5.5 Summary 119

Concerning the distinction in the contact area between these two entities with

different intrinsic structures, as mentioned in the above paragraph. The Figure

5.11(b) shows that a distinguishable variance of the predicted dimensionless

contact radius (the radius of contact area normalised by the particle radius),

a* , starts from a value of the dimensionless approach of 50 %. However, like

in the homogeneous elastomeric sphere, the measurement of the contact area

for the microcapsules has also been shown to be extremely difficult due to the

diffraction limit of the optical system (about 1 urn). In the current case, for

a 65 urn microcapsule, at 20 % deformation, the radius of the contact area

is about 16 im according to the theoretical prediction (Figure 5.10). Hence,

the resolution of optical system would result in an error of possible as large as

about 6 %. Yoneda (1973) has attempted to directly measure the contact area

of compressed sea urchin eggs (ca 120 tim) between two plates by the use of

photomicrographs taken from the side view. He pointed out that it would be

nearly impossible to detect a very narrow gap between the egg and the plate,

using optical methods, and this would cause a major over-estimation of the area

of contact. Of course, these measurements may be accurately resolved by a new

method; for example, X ray imaging is a possibility.

5.5 Summary

A model for a liquid-filled spherical membrane model with either a Mooney-

Rivlin law or a Neo-Hookean constitutive equation has been verified to be valid

for the compressive deformation behaviour of a single microcapsule. This model

allows correlations to be made using the experimental micro-compression data

for the determination of the elasticity and the tension distribution and bursting

strength of the membrane as well as the pressure difference across the mem-

5.5 Summary 120

brane. The feature where the internal pressure nonlinearly increases with the

compressive deformation may be an extremely important result for the usage

of microcapsules as drug, or active species, delivery systems. The characteri-

sation of the major geometric deformational parameters has been proven to be

important, but difficult, for the exploration of the physical properties and the

confirmation of the theoretical model.

Chapter 6

The deformation of biologicalcells

6.1 Introduction

The deformation of biological cells which is mainly governed by their mechan-

ical/ viscoelastic properties , as well as the bursting strength, may play an

important role in many cellular processes. It may be, in some circumstances,

that mass transfer effects may be of significance. For example, in context of physi-

ological processes, the mechanical and viscoelastic properties of plant cell walls

regulate the expansion in response to the Turgor pressures , allowing the plant

cell to enlarge by an order of magnitude or more. Without this mechanism,

the growth and morphogenesis of the plant would be impossible (Niklas 1992).

Regarding the bursting strength, in many cellular engineering processes, the

rheology is changed significantly due to the formation of broken cells. For ex-

t Some experiments (Fry e al. 1992) have shown that the potential mechanisms for regu-

lating the cell wall expansion is via specific enzyme action. However, in the current study we

only focus on the purely mechanical! viscoelastic aspects.tWhen matrix potentials are neglected, the Turgor pressure is defined as the difference

between the water chemical potential and the solute chemical potential of a cell or tissue.

The Turgor pressure is biomechanically important because it profoundly influences the tensilestresses generated within cell walls and hence the mechanical stiffness of thin-wall cells and

tissues.

121

6.1 Introduction 122

ample, in commercial food pastes, the cell ghosts or fragments no longer behave

as deformable particles showing Hertzian behaviour . In order to understand

the biological cell deformation, a concise description of the constitution of both

plant and animal cells is appropriate and this is given in the following paragraph.

Eukarytotic cells have at least four major structural domains: A plasma

membrane to define the boundaries of the cell and retain its contents; a nucleus

to house its DNA, membrane-bounded, organelles; and the cytoplasm with its

cytoskeleton of tubules and filaments. In addition, plant cells almost always have

a rigid cell wall, and animal cells frequently have a cell coating of glycoproteins

(Becker 1986). Most of biologists agree that the cell wall for plant cells and the

plasma membrane for animal cells are important to maintain the cell integrity.

However, from the physical point of view, a membrane is not simply a lipid

bilayer studded with protein as many would biologists suggest. A better defi-

nition is: "a material with a very small thickness, in comparison with its radii

of curvature, which separates two adjacent liquid-like domains and supports the

stress created by the embedding medium" (Fisher 1993). From this point of

view, a single plant cell may be modelled as a liquid-filled spherical membrane

structure. Two typical types cells are shown in Figure 6.1.

There are some models, based upon elasticity theories, that have been pro-

posed in order to describe the interrelationship between stress and strain of

naturally occurring cell membrane systems. Skalak et al. (1973) were the

first to propose the general constitutive equations, based on the strain-energy

function approach, for describing the large scale deformation of biological cells.

Then, Hettiaratchi & O'Callaghan (1974; 1978) modelled the cell wall as a flex-

ible, long-chain polymeric material whose stress-deformation behaviour could

This phenomenon has been noted by Professor P. J. Liliford e al. (Unilever, Colworth)

in their unpublished experimental data.

6.1 Introduction 123

CNamow -

IrI.'t •''/ J•

PT. ------4V .c' zr

C-

(a) GoIg,coClcoaw,

meane/ N snieop.

NiOk NucJe.s

/

A7L

s :. :1GO

Fi.s -'roomes

Figure 6.1: Schematic representations of typical structures of cells: (a) plant cell(b) animal cell.

6.2 Experimental 124

be characterised by a Mooney-Rivlin or a neo-Hookean substance. However,

the Mooney-Rivlin material law and also the neo-Hookean law both have been

proven to be only to be suitable for rubber-like material which may be consid-

ered as homogeneous hyperelastic materials. The cell wall is mainly composed

of many cellulose microfibrile embedded within a matrix. The contributions

of the microfibrils in maintaining cell wall integrity, that is restraining cell ex-

pansion, and regulating the direction of the expansion have been highlighted in

several studies (Wu et al. 1988; Chaplain 1992). In these studies the deforma-

tion of the cell wall has been assumed to be mainly regulated by two phases:

the relaxed-wall phase and the stressed-wall phase. The latter, moreover, has

been further subdivided into two phases known as the matrix-regulated phase

and the microfibril-regulated phase, respectively. Each phase is a mechanically

distinct component and is separated by tension points which serve as useful ref-

erence points for the construction of mathematical models. Essentially, the wall

is considered to be a complex composite material.

6.2 Experimental

6.2.1 Material

Tomato cells were isolated by soaking pieces of pericarp tissue in O.05M CDTA

(calcium chelator), Murashige and Skoog salt buffer solution, under mild ag-

itation. A very high percentage of cells were shown to retain their viability.

The shape of these cells was observed to be quasi-spherical (see Figure 6.2).

The diameter of the particles varied between 300 to 500 pm. These materials

were partly provided by Unilever Colworth and partly prepared by the author

according the rec ipe supplied by the Company.

6.2 Experimental

125

93-10-22 FRI2:9' :S4\

""?

I

(a)

I 0 - 2254I9

:II

fl ____

fl

FII

(b)

Figure 6.2: Photographs of the single isolated tomato cells: (a) green cell (ca

250 im diameter) (b) red cell (ca 250 um diameter).

6.3 Theoretical analysis 126

6.2.2 Stress-relaxation experiment

After the initial contact point between the cell and a microplaten was found, the

cell was rapidly compressed by the microplaten driven at a constant speed, typ-

ically 100 zm/sec, to a predetermined deformation (or dimensionless approach).

Then the decay in the reaction force, at the fixed imposed deformation, was mea-

sured over time of upto 60 sec. The applied deformations used in the current

study were varied from 80 % to 95 %.

6.3 Theoretical analysis

6.3.1 Constitutive equations

6.3.1.1 General constitutive equations for cell membrane

Essentially, the constitutive equations, proposed by Skalak et al. (1973), have

included the unique feature, as observed in red blood cells (RBC) by numerous

experimental tests, that the cell membrane can sustain a large shear deformation

but only a small area change (the maximum change is about 7 % in the RBC).

Hence, contrary to the Mooney-Rivlin law in which the stresses of uniaxial ten-

sion and isotropict tension are assumed to be the same order of magnitude, in

this (Skalak et al.) strain energy function these two stresses are three orders of

magnitude different. The simple form of the strain energy, w;, described by the

Skalak model is;

w*- -(I+I1_I2) +I22(6.1)- 4h

where B and C are membrane material properties, assumed to be constants

in this study; h is the initial thickness of the wall material, I and '2 are the

tlsotropjc is used here to denote a state of the stress resultants in which T1 =T2 . The

definition of T1 and T2 are presented in a following paragraph of this section.

(6.2)

(6.3)

(6.4)

(6.5)

CA1 I - i) + AI2]

CA2 1A2 - i) + AI2]2

U1h

= A1A2(6.6)

(6.7)


stretch-ratio invariants which may be expressed in terms of the principal stretch

ratios in the meridional and circumferential direction, A and )'2, respectively,

and are shown as below:

Il = A + A —2

j - —12 - A1A2

Thus, only two elastic constants (B and C) are required in order to charac-

tense the platen induced, elastic deformation of the membrane. Then, the re-

lationships between the principal stresses in the meridional and circumferential

directions of the deformed surface can be expressed as below;

ÔW* - 62 (r (A - i) + AI2 )cT1A1

ôA 1 - 2h 1

2 (r (A - i) + AI2)U2 = A2 8A2 = 2h 2

where r = B/C. Thus, the stress resultants per unit length, T1 , T2 , can be

expressed as the simple functions of the principal stretch ratios, A 1 and A 2 (see

Figure 5.2), by;

The shear modulus (the constant B divided by the membrane thickness) has

been shown to be about three order smaller than the area modulus (the con-

stant C divided by the membrane thickness) in some experimental investigations

(Evans & Skalak 1979; Hochmuth 1987) . Thus r may be reasonably assumed

to be 0.001 in the current study.

Although the tomato cells that have been examined in the current study

may be slightly different from red blood cells, for which the model was originally

developed, the general deformation behaviours of the tomato cells and the RBC

have been proven to be quite similar.


6.3.1.2 Specific constitutive equations for plant cell

Recently, a set of constitutive equations has been developed by Chaplain (1992),

specifically for modelling the deformation of plant cells. This model is essentially

based on a general class of strain energy function which was originally proposed

by Ogden (1972) in relation to experimental work on rubber-like materials. This

strain energy function has been successfully applied for modelling the cell wall

of acetabularia (Chaplain & Sleemanl99O). The relationship between the strain

energy function, W, and the principal stretch-ratios has been formulated as;

n

w: = /irq(cr) (6.8)r1

where

q(c) = (.' + + ) - 3)/& (6.9)

and ,. and cr,. are real constants with the condition that; /i T cx > 0 for each r

(no summation).

It can be shown that the following relationship holds between the elastic

shear modulus, jt, and the parameters j and ct,.

n

= >/1rclr (6.10)1=1

A nonlinear relationship between the Turgor pressure and volume expansion

of plant cells has been reported by Gardner and Ehlig (1965) through exper-

imental investigation. In order to describe this phenomenon, a "two term"


strain energy function has been introduced by Chaplain (Chaplain 1992). The

function, W, is composed of two distinct components. One describes the char-

acteristics of the matrix-regulated phase and the other expresses the behaviour

of microfibril-regulated phase. It may be written as;

T147 = l'Vrnriz + VVTrnicrojibrils (6.11)

Hence, the deformation of the cell wall may be characterised by the "two term"

strain energy function along with the parameters p 1 ,c, /2, cr 2 . It is obvious that

this strain energy function provides more physical insight than the previous one

proposed by Skalak et a!., especially for the deformation of plant cells. However,

the extra constants which are involved in this type of strain energy function

are difficult to determine. Hence, the application of this function is impractical.

For this reason, only the constitutive equations proposed by Skalak et al., are

adopted in the subsequent analyses described in the present work.

6.3.2 Governing equations for compressive compliance

Many studies (Fung 1966; Fung & Tong 1968; Skalak et al. 1973) have modelled

the quasi-spherical cell as an spherical elastic membrane which is filled with an

incompressible viscous fluid. Hence, the membrane model, as mentioned in the

Chapter 5, may be applied for describing the deformation of the tomato cell. In

this case the governing equations for the deformation of the cells are the same

as the equations (5.11) to (5.15). However, due to the nature of the constitutive

equations for biological membrane which are adopted, the variables, f, f2 and

f are included in the equations and are reformulated as below;

ôT1 = -- 2rA 1 + +11 = ;5i;-


± 1 - i) + A 2 (A 2 A 2 - i)] (6.12)2 124A 2 I8T1 - A1 I2A2A3 + 2A2 (A2A2 - 1)] -f2

—L12 12

A1A - i) + A 2 fA2A2 - i)} (6.13)

4A2L \ 21 2

C A3 )3\

1 213 = Tl_T2=(__-_) (6.14)

6.3.3 Stress Relaxation analyses

Mechanical models are widely applied to describe the viscoelastic behaviour of

materials. In Figure 6.3 are shown three mechanical models which are widely

used for describing the viscoelastic deformation of materials, namely, the Maxwell

model, the Voigt model and the Kelvin model (or Standard Linear Solid). Essen -

tially, all of these models are composed of combinations of linear springs with a

spring constant, i, and dashpots with coefficient of viscosity, A linear spring

is supposed to produce instantaneously, a deformation which is proportional to

the load. A dashpot is supposed to produce a velocity proportional to the load

at any instant. Thus, if F is force acting in a spring and u is its extension, then

F = u. If the force acts on a dashpot, it will produce a velocity of deflection

ü, and F =

It is intuitively clear that for the Maxwell solid, a sudden application of

a deformation to the model material produces an immediate reaction by the

spring, and then a stress relaxation follows. The relaxation is provided by the

response to the deformation of the dashpot in which the displacement increases

linearly with time. Eventually, the reaction force will exponentially decay to

zero due to the complete relaxation of the dashpot (see Figure 6.3). For the

Voigt case, a sudden application of a deformation will instantaneously produce,

in theory, a infinite reaction force, due to the resistance contributed by the

dashpot. Then, the applied force will be shared by the spring and maintained


(a) A Maxwell body

1'l

I i

iaiij/2Si

Fli p ______ ____S MW •4 S ii

II

Ii iii

I P 5 P 5

U 1 w U'1_'I

(b) A Voigt body

_____ F1

F 'F____ F

(c) A Kelvin body (a standard linear solid)

/21 F1

fF sW1v I

FUI I'll U'i ill

Po F0

U_pri

Figure 6.3: Three mechanical models for a viscoelastic material. (a) A Maxwell

body, (b) a Voigt body, and (c) a Kelvin body (a standard linear solid). Adopted

from Fung (1993) with permission.


constant as time lapses (again, see Figure 6.3). The characteristics of the stress

relaxation for these two models are obviously different from those observed in

the present experimental results shown in Figure 6.5.

For modelling the experimental data, shown in Figure 6.5, a Standard Linear

Solid model (or Kelvin model) has been employed to interpret the experimen-

tal stress-relaxation curves for both the green and the red tomato cells. The

characteristics of the stress relaxation for the Kelvin material (Figure 6.5) may

be simply described by the following mathematical formulation (the detailed

derivation of this equation is given by (Fung 1993));

= [i - (i - I\ e 1 (6.15)r) J

where the constant r represents the factor, ij/p, which is the relaxation time for

a constant strain; r is the relaxation time for a constant stress. The parameter

r is the characteristic time of the relaxation of the load under the condition of

a constant deflection, whereas the quantity r, is the characteristic time of the

relaxation of the quantity deflection under the condition of a constant load. F

is the reaction force which is a function of time, t, and F0 is the initial reaction

force at t=O.


6.4.1 Stress relaxation

Typical experimental results for the stress relaxation process for both the green

and the red tomato cells are shown in Figure 6.5. The theoretical correlation

for the experimental data, using the equation 6.15, is also shown in the Figure

6.5. The relaxation time for a constant strain (ca 85 %) can be calculated to

be 1.36 s for the green cell and 1.90 s for the red cell. The relaxation for a

Time

(0) (b)

a)0I-0U-

00)a

a)00

0a)a

0)00

E00)a


Time

(c)

Figure 6.4: Relaxation behaviours of (a) a Maxwell, (b) a Voigt, and (c) a

standard linear solid.


constant stress can is calculated to be 1.94 s for the green cell and 7.38 s for

the red cell. These small values of the relaxation constants are consistent with

the rapid decay of the stress (or strain) under a constant imposed strain (or

stress), as observed in the experimental data (see Figure 6.5). These results also

imply that the deformation behaviour of the tomato cells appears to be more

viscous-like as opposed to elastic-like in their nature. This peculiar feature may

be due to the rupture of the plasma membrane which often occurs at very small

imposed strains. Once the plasma membrane is broken, the cell wall which is

now a highly porous material, allows the contained fluids to be readily transport

out of the cell. Hence, it may be concluded that this relaxation phenomenon

seen in the tomato cells is partly contributed by the transport of the contained

fluids.

6.4.2 Compressive compliance and bursting phenomena

The typical experimental curves, describing the force change against the dis-

placement, for both red and green tomato cells are shown in Figure 6.6. The

tomato cell, unlike the liquid-filled microcapsules (Chapter 4), can initially re-

sist large deformations without a significant reaction force increase. When the

deformation reaches certain point, the force starts to increase rapidly until the

cell bursts. The bursting behaviours of the two cells are different. The green

cells have very clear bursting points, whilst, the red cells do not show such clear

rupture points (again, see Figure 6.6). This phenomena may imply that the cell

walls of the green cells are rather brittle whereas those of the red cells are more

ductile.

The theoretical predictions of compressive compliance, based upon an ex-

tension of the Skalak model, are shown in Figure 6.7. It is clear that the char-

acteristics of the theoretical predictions are inconsistent with the experimental

1

0.9 -

0.8

0.7 -

0.6

0.5

0.4

0.3

0.2

0.1

0

Experimental data

- Kelvin model


0 10 20 30 40 50 60 a

Time (sec)

(a)

1.

0.9 -- Experimental data

0.8-t Kelvin model0.7 -

0.600.5

0.2 ---

0.1

0

0 10 20 30 40 50 60

Time (sec)

(b)

Figure 6.5: The experimental results and theoretical correlation of the stress

relaxation for two types of tomato cells (a) green cell (b) red cell.

2.2

2

1.8

1.6

t


1.7

1.6

1.5

1.4

1.3

1.2

1.1

lii0.9

2 0.8

0.7

06

0.5

0.4

0.3

0.2

0.1

00 40 80 120 160 200 240

DISTANCE(M ICRONS)

(a)

0 200 400 500

DISTANCE(MICRONS>

(b)

Figure 6.6: Typical experimental curves of the compressive compliance for two

types of tomato cells; (a) green cell (b) red cell.

(5.4 R'suIts a 11(1 (Ii.SUIISSIOIIS 137

40 I I I

30 - -

20 - -

10 - -

0 -

—10 I I I I

—0.2 0.0 0.2 0.4 0.6 0.8

1— (/r0X)

Figure 6.7: rFhe theoretical predictions for the compressive compliance for the

spherical cells, obtained by using the membrane model combined with the se

lected constitutive equations for the cell membrane.


results shown in Figure 6.6. This inconsistency may arise from various possible

error sources. Firstly, as mentioned in the section 6.4.1, the plasma membrane

may rupture at small imposed strain. Then, the enclosed volume provided by

the cell walls may not remain constant, due to the lost of the contained fluid.

Furthermore, up to now there is no study that has examined quantitatively the

resistance to deformation contributed by the cytoskeleton (Petersen et al. 1981;

Evans & Skalak 1979). However, this part of the cells may play a significant

role in the generation of the reaction force at the large deformation. Finally,

the constitutive equations proposed by Skalak et al. (Skalak et al. 1973), which

have some deficiencies in the consideration of the specific deformation mecha-

nism of the cell walls may cause the theoretical model fail to accurately predict

the compliance curve.

Chapter 7

Conclusions

The experimental techniques, combined with the appropriate analyses, described

in the Thesis have investigated the deformabilities of single micro-cellular enti-

ties. This has included both the measurement of the load-displacement response

curve and the deformed shapes. Using the available theories, some of the impor-

tant mechanical and interfacial properties have been estimated. These param-

eters, as reported in the previous Chapters, include the Young's modulus, the

Poisson ratios and the interfacial free energy for the homogeneous elastomeric

spheres. For the liquid-filled elastomeric membrane systems, the elasticity of the

membrane, the tension forces on the membrane, the bursting strength and the

internal pressure have been calculated. For the biological cells (tomato cells),

due to the complex intrinsic structure and the peculiar material properties, the

mechanical and interfacial properties have proven difficult to determine. The

main specific conclusions of this study may be summarised in the following ma-

jor parts; (1) the governing effects responsible for the observed deformation,

(2) the experimental method and (3) the theoretical interpretations. They are

summarised in the following sections.

139


7.1 The governing effects responsible for theobserved deformation

There are some important effects which govern the deformation of micro-cellular

entities. These effects have been examined in the current study and reported in

the previous Chapters. Although these effects, which are mentioned below, may

be specified as unique mechanisms, most of them are often interrelated to each

other in the practical cases.

1. The Autoadhesion: The effect has been shown to be extremely important

in the small deformation range and even in the medium deformation re-

gion (dimensionless approach upto 25 %) for the homogeneous elastomeric

sphere in dry contacts. The influence of this effect has been quantitatively

described by the JKR theory (Johnson ci al. 1971). Since, for the liquid-

filled membrane system and the biological cells, these investigated entities

must be surrounded in the liquid medium for practical reasons, the au-

toadhesion is negligible. However, for the case of the dry contact in these

systems, the influence still awaits a detailed examination. This autoad-

hesion effect is also influenced by the platen roughness, the viscoelastic

effects and the scaling effect (see the following paragraphs).

2. Large deformations: For the case of homogeneous elastomeric spheres,

the large deformation effect t has been shown to be important when the

ratio of the radius of the contact area to the radius, a/R, achieves a value

of approximately 0.3. For the liquid-filled membrane systems, since large

contact areas are easier to achieve than for the homogeneous elastomeric

spheres, the effect becomes relatively more important. However, this effect

usually is linked with the non-linear elasticity of the system.

tThe definition of "large deformation" has been reported in Section 4.2.1.2.


3. Non-linear elasticity: This effect has been shown to be important for

elastomeric materials at larger deformations. Based upon the Mooney-

Rivlin law, the formulations of non-linear elasticity have been given by

Tatara, for the homogeneous elastomeric sphere, and by Feng and Yang,

for the liquid-filled membrane system. These formulation, in principle,

quantitatively introduce the effect of the elasticity increasing with the

strain. Furthermore, for biological cells, this effect has also been shown

to be extremely significant through experimental investigations. However,

the details of the formulation of non-linear elasticity, based on the strain-

energy function model, for the wall of the plant cell still awaits resolution.

4. The roughness: The influence of the roughness, both of the platens and

the particle surfaces, on the investigated micro-cellular entities may be im-

portant in two main respects; the reduction of autoadhesive force and the

modification of the friction force. The former aspect has been examined,

based upon the work of Fuller and Tabor (1975), both experimentally and

theoretically in the current study. The autoadhesive force has been sig-

nificantly reduced by the roughness of the platens for the homogeneous

elastomeric spheres.

5. Viscoelastic effects: This effect plays an important role on both the au-

toadhesion and the bulk mechanical responses. In the current work, the

measured adhesive force between the elastomeric micro-sphere and the

platens has been found to slightly increase with the increase in the rate

separation and this phenomenon may be contributed to the viscoelastic

properties of the elastomer (Greenwood & Johnson 1981). Concerning the

bulk mechanical responses, the influence of the viscoelastic effect has been

shown to be very important for the biological cells, but to be rather neg-

7.2 The experimental method

142

ligible for both the homogeneous elastomeric spheres and the liquid-filled

membrane systems within the time scales examined.

6. The scaling effect: In general, when the size of the micro-cellular entities

decreases, the influence of the autoadhesive force becomes more important.

At very low applied loads the magnitude of the adhesion force, compared

with that of the mechanical force proposed by Hertz, has a greater contri-

bution to the deformation of elastomeric micro-spheres; the compressive

force is approximately equal to the square of the radius at a constant

imposed strain, while the adhesion force is directly proportional to the

radius.

7.2 The experimental method

1. The principle of instrument: Based upon the conventional compression

method, the current method (micro-upsetting) includes some of the main

elements from both the conventional imaging method and the compliance

technique adopted in nano-/ micro-indentation studies. It has been shown

to provide, in principle, the required information about the deformation of

the micro-cellular entities. Furthermore, the current method offers more

measurable parameters than most of the previous methods which is crucial

for the verification of the theoretical modelling.

2. Initial contact point: This technical problem has been recognised to be

intractable in the many previous studies on the nano-/ micro-indentation.

For the current study, an uncertainty of approximate 0.6 im has been

generally found within the various experiments. For very small micro-

cellular entities, this uncertainty may cause serious error in the prescription

of the load-displacement curve. However, this uncertainty seems to be

7.3 The theoretical interpretation 143

tolerable for the investigated micro-cellular entities adopted in the current

work (which are larger than 65 gm).

7.3 The theoretical interpretation

7.3.1 For the homogeneous elastomeric spheres

1. The theories without autoadhesion effects: The Hertz theory has been

shown to be inadequate in order to describe the deformation at both very

small and large deformations due to the lack of the consideration of the

effects of the autoadhesion as well as large deformation. The Tatara the-

ory which includes both the effects of the non-linear elasticity and the

large deformation has been shown to be appropriate for the cases of the

dimensionless approach upto about 40 %. However, the neglect of the au-

toadhesion in the theory may cause the errors to some, but an unknown,

extent in the prediction of the deformation.

2. The JKR theory: The theory has been shown to be both correct and

efficient for the description of the deformations which involve the autoad-

hesive force at small compressive deformations. Furthermore, the theory

also provides an useful means to estimate the interfacial properties through

the investigation of pull-off force.

7.3.2 For the liquid-filled membrane

1. Membrane model: The adopted model, associated with either the Mooney-

Rivlin law or the Neo-Hookean constitutive equations, gives the good pre-

dictions of the loading! unloading curve as well as the corresponding ma-

jor geometric deformational parameters for the liquid-filled microcapsules.

7.3 The theoretical interpretation

144

However, the model fails to predict the deformation of the plant cell pos-

sibly due to the selection of inappropriate constitutive equations.

Appendix A

The jump phenomenon

A.1 Long range forces: Lifshitz theory

Tabor & Winterton (1969) were first to apply the Lifshitz theory (1956) to study

the characteristics of jump contact between the two orthogonal cylindrical sheets

of mica. The force, F, between a sphere with a radius, R, and a infinite flat

plane at a separation, H, can be expressed as;

for non-retardation forces;AR

F 6H2

for retarded forces;- 2irBR

F 3H3

(A.1)

(A.2)

where A is the non-retarded Hamaker constant and B the retarded Hamaker

constant (Bradley 1932; de Boer 1936; Hamaker 1937).

Israelachvili & Tabor (1972) showed that the essential equations (A.1) &

(A.2) could be derived using much simpler theoretical approaches than originally

adopted by Lifshitz (Tabor & Winterton 1969; Israelachvili & Tabor 1972).

The detail derivations of the equations of Lifshitz theory are omitted in this

appendix. However, the final equations of Lifshitz theory for the calculations of

the Hamaker constant and the retarded van der Waals constant may be expressed

145

A.2 The experimental configuration and the origin of the "jump" 146

as follows;

A= f'de (A.3)4irJo

where2 Wf (w) dw

(A.4)R0 C.i)

and12

B = - ) c5 (Eo) (A.5)

+ 1

where h is Planck's constant; is the electrostatic dielectric constant; is the

dielectric constant (or permittivity); E" is the imaginary part of the dielectric

constant of the material of the plates and is a function of angular frequency, w;

c is the speed of light in vacuum. Values of (e 0 ) have been given by Lifshitz.

For o less than about 3, (to) is 0.35.

A.2 The experimental configuration and theorigin of the "jump"

Tabor & Winterton (Tabor & Winterton 1969) used a flexible cantilever to

support one surface; the other was attached to a piezoelectric microtranslation

stage. The configuration has been widely used by Israelachvili, Luckham and

others. (for reviews see Israelachvili (1991); Luckham & Costello (1993)). A

similar configuration is applied in the current work.

As the surfaces approach the increasing van der Waals attractive force pulls

the surfaces together to a new equilibrium position where the spring deflection

is x. The spring stiffness of the cantilever beam is a constant, K. Hence,

the van der Waals attraction force, for the case of the non-retarded forces, is

balanced by the restoring force of the spring of the cantilever beam, Kx, so that

at equilibrium we have

A.3 The Johnson method 147

F=Kx— (A.6)

where F is net force. The jump occurs when the gradients of the van der Waals

and spring forces are balanced, that is when

dF = Kdx + 2--dH =0 (A.7)6H3

Since dx = —dH this implies that the surfaces will jump into contact when

A/3H = K/R (A.8)

where H0 is the jump distance. A similar analysis can be applied for the case

of the retarded forces but the necessary calculation is within the non-retarded

range.

The computed jump distance based upon a non-retarded force is recorded in

Section 4.4.1.

A.3 The Johnson method

The author is grateful to Prof. K. L. Johnson (Cambridge) for privately com-

municating the following analysis. It should be noted first that the value of, H0,

obtained by Winterton was directly measured using fringes of equal chromatic

order (FECO) optics. In the present experiments it has to be inferred from the

displacement as sensed by the force transducer after the jump O-E in figure

A.1(a). The following analysis provided by Professor Johnson deals with this

problem.

The figure A.1(a) shows the JKR compressive load (P)-defiection (ce) curve.

The point P, as shown in Figure A.1(a), corresponds to the pull-off point with

an infinite stiff dynamometer spring. The point Q is the pull-off point with a

At point Q

(i) At point D

(ii)

A.3 The Johnson method

Tension(a)

148

(b)

Figure A.1: (a) The compressive load (P)-deflection (&) produced by the JKR

theory for the interplation of the jump phenomenon (Johnson 1995; private

communication). (b) The actions of the sphere and platen at (i) the point Q

and (ii) the point D

A.3 The Johnson method 149

dead load (zero stiffness spring). The point R is the pull-off point for a typical

finite stiffness spring. Starting with the sphere in contact with the lower platen.

The initial condition is shown in figure A1.b(i). Neglecting the weight of

the sphere (P=O) the lower contact condition is given by the point A on the

P-cr curve. The platens approach slowly and steadily with the gap at the upper

contact point is within the range of surface forces. This is the jump distance H0

in equation (A.8). It can be neglected so that the upper contact is effectively

made at point 0 in the Figure A.1(a).

The initial "jump" at this point of contact is from point 0 to E and takes

place at the Rayleigh wave speed for the material of the sphere. The sphere is

now not at equilibrium under the action of PE at the upper point of contact and

zero at the lower. It moves to make the (tensile) forces equal at point D, along

ED, in the Figures A.1(a) & (b).

During this movement the centre of the sphere displaces upwards by AB=BC.

The extension of the dynamometer spring by load PD is the quantity OC. The

movement of this platen in this interval of time is negligible. The perceived

jump distance deduced from the equilibrium displacement of the spring is thus

the value of OB which from Figure 4.5 is about 500 nm.

Appendix B

Computer programs

150

Computer programs 151

C***********************************************************************cC THIS PROGRAM IS DEVELPED FOR THE MODELING OF THE LARGE DEFORMATION CC OF RUBBER ELASTIC SPHERE 21/06/93 KKL CC***********************************************************************C

program rubberreal*8 P,GAMAL,GAMAJI,GAMAH,FL,FM,FH,A,AL,AH,AM,gama,UrREAL*8 APLUM,E0 ,ETA,POI ,Z,R,pi ,gamahz ,ahzREAL*8 F40,F31,UZA,ACOF,BCOF,FUNA,FUNAPLUcommon

1/var/P ,GAMAL,GAMAM,GAMAH, FL, FM, FH ,AL,AH,AM,2 APLUM,E0,ETA,POI,Z,R,pi,3 F40, F31 ,UZA,ACOF,BCOF,FUNA,FUNAPLUOPEN (5,FILE='TAtar.dat')open (6,file='rubber.out')poi=0 .50E0=5. 0e9R=205E-4write (6,400)

400 format(/,' Poisson Y-modulus radius')write (6,500) poi,E0,R

500 format (3(flO.3))write (6,200)

200 format(/,' P GAMA a* a z U(z,a) U ( R)1 ratio')write (6,300)

300 format(' (N) (cm) (cm) (cm) (cm) (cm)

(cm)')ncont=2 2n=0itermax=100iter=0pi=3 .141592654

40 READ(5,*) pn=n+1DO 1 J=1,199gama=O .0a=0 .0f40=0.0iter=iter+1GAMAL=102 . 5e-6*(J-1)CALL F43s(GAMAL)FL=F40GAMAH=GAMAL-t-102. 5e-6CALL F43s(GAMAH)FH=F40IF ((FL*FH).Ge.0.0) GO TO 10

20 GAMAN=(GAMAL*FH_GAMAH*FL)/(FH.-FL)CALL F43s(GAMAM)fm=f 40if (iter.gt.itermax) go to 30IF (abs(GAMAh-GAMAL).LT.0.1E-6) GO TO 30IF ((FM*FL).gT.0.0) THENFL=FMGAMAL=GAMANELSEFH=FMGAMAH=GAMAMENDIFGO TO 20

10 continuecontinue


H

30 Ur=ACOF*(1.0+POI)*P/(pi*E0)*(1.0+BCOF*Am**2/(5.0*R**2.0))*(1.0/(2.0**1.5)_(1.0_2*POi)*(1.0_1.O/(2.O**O.5)))

write (6,100) p, gamam, aplum, ,z , uza, urahz=((3.0*(1_poi**2)*R*p)/(4.0*E0))**0.333333gamahz=3 . 0*(1_poi**2)*p/(4*E0*ahz)ratio= (aplum/ahz ) **2 .0

100 format(/,2x,e7.1,2(e7.4) ,3(e7.3) ,e7.4,e7.4)if (n.eq.ncont) go to 1000go to 40

1000 stopENDSUBROUTINE F43s(GAHA)real*8 fha,fla,fmareal*8 P,GAJ4AL,GA14AI4,GANAH,FL,FM,FH,A,AL,AH,AN,gamaREAL*8 APLUM,E0,ETA,POI,Z,R,piREAL*8 F40,F31,UZA,ACOF,BCOF,FUNA,FUNAPLUcommon

1/var/P,GANAL,GAMAN,GAMAH,FL,FM,FH,AL,AH,AN,2 APLUM,E0,ETA,POI,Z,R,pi,3 F40,F31,UZA,ACOF,BCOF,FUNA,FUNAPLUitermax=100iter=0DO 1 1=1,19f31=0 .0iter=iter+1AL=(I_1)*1O . 25e-4+5. le-4CALL F31s(GAI4A,AL)FLa=F3 1AH=AL+10. 25e-4CALL F31s(GANA,AH)FHa=F3 1IF((FLa*FHa).GT.0.0) GO TO 10

20 AM=(AL*FHa-AH*FLa) /( FHa-FLa)cALL F31s(GAMA,AM)fma=f31if (iter.gt.itermax) go to 3if (abs(f31).le.1.Oe-13) go to 3IF (abs(Ah-AL).GE.1.OE-5) THEN

IF ((FMa*FLa).GT.0.0) THENFLa=FMaAL=AM

ELSEFHa=FMaAH=AM

ENDIFgo to 20

ELSEGO TO 3ENDIF

10 continueCONTINUEgo to 4

3 UZA=ACOF*(1.0+POI)*P/(2*Pi*E0)*(1.0+BC0F*Aifl**2/(5.0*1 R**2.0))*((Z**0.5+(2*R_Z)**0.5)/((2*R)**1.5)_2 (1.0_2.0*POI)/(2*R*Z*(2*R_Z))**0.5*(2*(2*R)**0.5_Z**0.54 _(2*R_Z)**0.5))

APLUm=am+UZAFUNAPLU=2*(1.0+pOI)*R**2/(APLUm**2+4.0*R**2)**1.5+(l.O_POT**2)/

1 (APLUm**2+4.0*r**2)**0.5F40_GAMA_3.0*(1.0_POI**2)*ACOf/(4*E0)*(1.0+BCOF*Am**2/(8*R**2


III

1 ))*p/APLUm+FUNAPLU*ACOF/(pj*EO)*(1.O+BCOF2 *Ait**2/(5.O*R**2))*P

4 RETURNendSUBROUTINE F31s(GANA,A)real*8 P,GAMAL,GANAM,GAMAH,FL,FM,FI-1,A,AL,AH,AN,gamaREAL*8 APLUM,EO,ETA,POI,Z,R,piREAL*8 F40,F31,UZA,ACOF,BCOF,FUNA,FUNAPLUCOMMON

, GAMAL , GAMAM , G,FL, FM, FH, AL,AH ,AM,2 APLUN,EO,ETA,POI,Z,R,pi,3 F40,F31,UZA,ACOF,BCOF,FUNA,FUNAPLUETA=GANA/RACOF= (1 . O-ETA)**2 . O/(1. O-ETA+ETA**2/3)BCOF= (1 . O-ETA/3 O)/(1 . O-ETA-f-ETA**2/3)FUNA=2*(1.O+POI)*R**2/(A**2+4.O*R**2)**1.5+(1.O_POI**2)/

1 (A**2+4.O*r**2)**O.5F31=R_(R**2.O_A**2.0)**O.5_ACOF*(3*(1.O_POI**2)/(8*EO*A)*(1.O-BCOF1 *A**2/(2*R**2.0))+BCOF*FURA*A**2/(2*pj*EO*R**2))*pz=r-(r**2-a**2) **O.5RETURNend

154Computer programs

* ****** * ** * ******* * * *

C This program is developed to caculate the contact problem of an inflated Cc spherical nonlinear membrane by KKL 01/11/94 Cc****************************** ********************************************* C

program Mainreal y(4),yp(4)REAL P0.THICK.R0,BE'rA.YPS.YPO,Y1CON.Y2CON.VO,VSTJMCOMMON1/VAR/ ALFA.BETA.ANGLE.YPS.YPO.Y1CON,Y2CON,V0,VSUM,R0,p,y3f, tf,y3i,2 INDF.Y1N(500),Y2N(500),y3N(500),ylC(500).y2C(500),y3C(500)3 TN(SOO).TC(500),PI,indc,indcf,yata(5oO),yataO,yatafy4fl(5o)4 ,tenl(SOO),ten2(500),tencl(500),tenc2(500),tl,t2open (5,file='tamat60.dat')P1=3 .1415962ALFA=0 .001'rHICK=O .02R0=1 .0yps=1 .0V0=4.0*PI/3 .0*R0**3BETA=THICK/R0pO=betayata0=0 .4do 50 n=1,500angle=0.2^(1.4-0.2) /500*nDO 10 1=1.400YPO=YPS+i*0 .0003 *ypwrite(6,*)(yp0 angle')write(6, *)ypo,angleCALL CONTAC1write(6,*) ('ylcon y2con')write(6, )ylcon,y2conDO 20 J=1,500P=(05+1O.0*j/500)*pO

c do 22 1=1,40C y3i=1.0-0.2/40n

CALL NONCAN1IF ( abs(y3f)At.O.04 .and. abs(yataf).lt.O.04

1 GO TO 1if (y3f.lt.-0.1 .or. yataf.lt.-0.1) go to 10write (6,*) ('yataf y3f')write(6,*) yataf,y3f

c 22 continue20 CONTINUE10 CONTINUE

go to 50write(6,*) ('boundary condition sati')write (6,*) ('yataf y3f')write(6,*) yataf,y3f

11 call inter(yatao,y2n,y4n,indf,vsum)30 CONTINUE

write (6,) ('voirat')write(6,*) vsum/vO

IF (abs(V0/Vsum-1.0) .1T.2.000E-2) thenGOTo 2else

go to 20end if

50 continuewrite(5,*) ('no solution')go to 99

2 WRITE(5,*)y VO VSUM angle')WRITE(5, *)vo.vs. anglewrite(5,*)(p ypO')write(5, )pypOwrite(5,8)(tc(i),ylc(i),y2c(i),tencl(i),tenc2(j)j1,jndcf)write(S,9)(tn(1),yln(i),y2n(j),y3n(i),y4n(j)tenl(j),ten2(j),

Compu ter programs 155

II

1 i=1,indf)8 format (3f8.5,16x,2f85)9 format (7f8.5)99 ER])

SUBROUTINE CONTACT CT,?, YP)REAL T,Y(4),YP(4),R,ALFACOMMONl/VARI ALFA, BETA,ANGL,E, YPS, YPO, Y1CON, Y2CON,V0.VSIJM, R0,p,y3f, tf,y3i,2 INDF,Y1N(500),Y2N(500),Y3N(500),Y1C(500),Y2C(500),Y3C(500),3 TN(500),TC(500),PI,indc,indcf,yaya(500),yata0,yataf.y4n(500),4 tenl(500),ten2(500),tencl(500).tenc2(500),tl,t2if (t.eg.0.0) thenyp (1) =0.0yp C 2) =0.0elseyt2=y(2)Tl=0.25*Y(1)*(alfa*(_1.+Y(l)**2)+Yt2**2*(_1.+Y(1)**2*Yt22))/Yt2F2=0.25*Yt2*(alfa*(_1.+Yt2**2)+Y(l)**2*(_1.+Y(1)**2*Yt2**2))/Y(l)f10.25*Y(1) * (2*alfa*Y(1)+2*Y(1) *Yt2**4) /Yt2 +1 0.25*(alfa*(_1.+ Y(1)**2) + Yt2**2*(_1.+Y(1)**2*Yt2**2))/Yt2f2=0.25*Y(l)*(2*Y(1)**2*Yt2**3^2*Yt2*(_1.+Y(1)**2*Yt2**2))/Yt2_

1 0.25*Y(1)*(alfa*(_l.+Y(1)**2)+Yt2**2*(_1.+Y(1)**2*Yt2**2))/Yt2**2f3=T1-T2f3=T1-T2

C fl=(1.0+alfa*yt2**2.0)*(1.0/yt2+3.0/y(l)**4.0/yt2**3.0)c f2=((_y(1)/yt2**2.0+3.0/y(l)3.0/yt2**4.0)*(1.0+alfa*yt2**2.0)+c 1 (y(1)/yt2_1.0/y(1)**3.0/yt2**3.0)*2.0*alfa*yt2)c tl=(y(1)/yt2_1.0/y(1)**3.0/yt2**3.0)*(1.0+alfa*yt2**2.0)c t2=(yt2fy(1)_1.0/y(1)**3.0/yt2**3.0)*(l.0+alfa*y(1)**2.0)C YP(1)=(1.0/((1.0/y(2)+3.0/(y(l)**4*y(2)**3))*(1.0+alfa*y(2)**2)c 1 )*((y(1)_y(2)*cos(t))/sin(t)*((y(1)/y(2)**2_3.0/(y(1)C 2 •*3*y(2)**4))*(10+alfa*y(2)**2)_2.0*alfa*y(2))+y(1)/c 3 y(2)/sin(t)*((y(2)/y(1)_y(1)/y(2))+alfa*(1.0/y(1)**3c 4 Iy(2)_1.0/y(1)/y(2)**3))))C YP(2)=( (y(l)-y(2) cos(t) ) /sin(t)c f3=tl-t2

yp(1)=_y(1)/yt2/sin(t)*(f3/fl)_(y(1)_yt2*cos(t))/sin(t)*(f2/fl)yp(2)=(y(1)-yt2cos(t))/sin(t)

end ifRRNER])

SUBROUTINE CONTAC1EXTERNAL CONTACTREAL T,Y(4),yp(4),TOUT,RELERR,ABSERRREAL TFINAL,TPRINT, ECC,ALFA,WORX(27)COMMON1/VAR/ ALFA,BETA,ANGLE,YPS,YPO,Y1CON,Y2CON,V0.VSUM,R0,p.y3f,tf,y3i,2 INDF,Y1N(500),Y2N(500),Y3N(500),Y1C(500).Y2C(500).Y3C(500),3 TN(500),TC(500),PI,indc,indcf,yata(500),yata0.yataf.y4n(500)4 •tenl(500).ten2(500),tencl(500),tenc2(500),tl,t2

INTEGER IWORI( (5) • IFLAG. NEQNNEQN=2T=0 .0Y (1) =yp0Y(2)=ypORELERR=1 . OE-7ABSERR=0 .0TFINAL=ANGLETPRINT=PI/100IFLAG=1TOUT=Tindc=0

10 CALL RKF4 5 (CONTACT. NEQN, Y. T, TOUT. RELERR. ABSERR, IFLAG. WORK. IWORK)ylcon=y (1)y2con y(2)


III

indc=indc+1tc(indc)=tylc(indc)y(1)y2c(indc)y(2)y3c(indc)y(3)tend (mdc) =tltenc2 (mdc) =t2indcf=indcGO TO (80,20,30.40,50,60,70.80), IFLAG

20 TOUT=T+TPRINTIF (Tout.LT.TFINAL) GO TO 10go to 222

30 WRITE(6,31) RELERR,AESERRGO TO 10

40 WRITE(6,41)GO TO 10

50 ABSERR=1.OE-8WRITE(6, 31) RELERR,ABSERRGO TO 10

60 RELERR=10 0*RELERRWRITE (6,31) RELERR, ABSERRIFLAG=2GO TO 10

70 WRITE(6.71)IFLAG=2GO TO 10

80 WRITE(6,81)11 FORNAT(F5.3.2F15.9)31 FORMAT(17H TOLERANCES RESET, 2E123)41 FORMAT(11H MANY STEPS)71 FORMAT(12H MUCH OUTPUT)81 FORMAT(14H IMPROPER CALL)

222 RETURN

SUBROUTINE NONCAN (I, Y, VP)REAL T.Y(4),YP(4).R,ALFACOMMON1/VAR! ALFA,BETA,ANGLE,YPS,YPO,Y1CON.Y2CON,V0,VSUM.R0,P,Y3f,tf,Y31.

2 INDF,Y1N(500),Y2N(500),Y3N(500),Y1C(500),Y2C(500),Y3C(500),

3 TN(500).TC(500),PI.indc,indcf,yata(500).yata0,yataf,y4fl(500)

4 ,tenl(500),ten2(500),tencl(500),tenc2(500),tl,t2yt2=y(2) /sin(t)T1=0.25*Y(1)*(alfa*(_1.+Y(1)**2)+Yt2**2*(_1.+Y(1)**2*Yt2*2))/Yt2T2=0.25*Yt2*(alfa*(_1.+Yt2**2)+Y(1)**2*(_1.+Y(1)**2*Yt2**2))/Y(1)fl=025*Y(1)*(2*alfa*Y(1)+2*Y(1)*Yt2**4)/Yt2 +

1 0.25*(alfa*(_1.+Y(1)**2)+Yt2*2*(_1.+Y(1)**2*Yt2**2))/Yt2f2=0.25*Y(1)*(2*Y(1)**2*Yt2**3+2*Yt2*(_1.+Y(1)**2*Yt2**2))/Yt2_

1 0.25Y(1) * (alfa* (-1.+Y(1) **2)+Yt2**2* (-1.+Y(1) **2*Yt2**2) ) /Yt2**2f3=T1-T2

c fl=(1.0+alfa*yt2**2.0)*(1.0/yt2+3.0/y(1)**4.O/yt2**3.0)cC 1 (y(1)!yt2_1.0/y(1)**3.0!yt2**3.0)*2.0*alfa*yt2)c tl=(y(1)/yt2_1.0/y(1)**3.0!yt2**3.0)*(1.0+alfa*yt2**2.0)c t2=(yt2/y(1)_101y(1)**3.0/yt2**3.0)*(1.0falfa*y(1)**2.0)c f3=tl-t2

yp(1)=(y(2)*cos(t)_y(3)*sin(t))/sin(t)**2.0*(f2/fl)_(y(3)!Y(2))

1 *(f3/fl)

yp (2) =y (3)yp(3)=yp(1)*y(3)Iy(1)+(y(1)**2.O_y(3)**2.0)/Y(2)*(t2/tl)_Y(l)/tl*

1 (y(1)**2.0_y(3)**2.0)**0.5*p/2.0/betaif (t.eg.angle) thenyp(4)=_rO*y3i*y(1)elseyp(4)=_r0*(y(1)2.0_yp(2)**2.0)**O.5end ifRETURN

Coinpu ter programs 157

Iv

SUBROUTINE NONCAN1EXTERNAL NONCANREAL T, Y (4) • TOUT, RELERR, ABSERRREAL TFINAL,TPRINT, ECC.ALPA,WORK(27)INTEGER IWORK (5) IFLAG, NEQNCOMMON1/VAR/ ALFA,BEI'A,ANGLE,YPS,YPO,Y1CON,Y2CON,V0,VSUM,R0,p,y3f, tf,y3i,2 INDF,Y1N(500),Y2H(500),Y3N(500),Y1C(500),Y2C(500),Y3C(500),3 TN(500),'FC(500),PI,indc,indcf,yata(500),yata0,yataf,y4rj(500)4 ,tenl(500),ten2(500),tencl(500),tenc2(500),tl,t2NEQN=4T=ANGLEY(1)=Y1CONY(2)=SIN (ANGLE) Y2CONY(3)=Y1CON* 0.99y(4)=yata0RELERR=1 0E-7ABSERR=0 .0TFINAL=PI/2 .0TPRINT=PI/200IFLAG=].TOUT=TIND=0

10 CALL RKF4S (NONCAn, NEQN, Y, T, TOUT, RELERR, ABSERR, IFLAG, WORK, IWORK)y3i=y3fy3f-y(3)tf=tyatat=y(4)IND= 1+ INDY1N(IND)=Y(1)Y2N(IND)=Y(2)Y3N(IND)Y(3)y4n(ind)=y(4)teni (md) =tlten2(ind)=t2tn(ind)=tINDF INDGO TO (80,20,30,40,50,60,70,80), IFLAG

20 TOUT=T+TPRINTTOLD=TIF (Tout.LT.TFINAL) GO TO 10go to 111

30 WRITE(6,31) RELERR,ABSERRGO TO 10

40 WRITE(6,41)GO TO 10

50 ABSERR=1.OE-8WRITE (6,31) RELERR, ABSERRGO TO 10

60 RELERR=10. ORELERRWRITE(6, 31) RELERR,ABSERRIFLAG=2GO TO 10

70 WRITE(6,71)IFLAG=2GO TO 10

80 WRITE(6,81)11 FORNAT(F5.3,3F15.9)31 FOR1IAT ( 17H TOLERANCES RESET, 2E12.3)41 FORNAT(11R MANY STEPS)71 FORMAT(12H MUCH OUTPUT)81 FORNAT(14H IMPROPER CALL)

222 continue111 RETURN


V

ENDSUBROU'rINE RKF4S (F, NEQN, Y. T, TOUT, RELERR. ABSERR. IFLAG. WORK, IWORK)INTEGER NEQN. IFLAG, IWORK(5)REAL Y(NEQN) ,T,TOUT,RELERR,ABSERR,WORX(27)EXTERNAL FINTEGER K1,K2,K3,K4,K5,K6,K1MK1M=NEQN+ 1K1=K1M+ 1K2=K1+NEQNK3 =K2 +NEQNK4=K3+NEQNK5=K4+NEQNK6=K5+NEQNCALL RKFS(F,NEQN,Y,T,TOUT,RELERR,ABSERR, IFLAG,WORK(1) ,WORK(K1M).

WORK(K1),WORX(K2),WORK(K3),WORK(K4),WORK(K5),WORX(K6).2 WORK(K6+1),IWORK(1),IWORK(2),IWORX(3),IWORK(4).IWORX(5))RFURNENDSUBROUTINE RKFS(F,NEQN,Y, T,TOU'I', RELERR.ABSERR. IFLAG,YP,H,F1, F2,F3,

F4, F5, SAVRE. SAVAE,NFE. KOP. INIT,JFLAG,KFLAG)LOGICAL HPAILD. OUTPUTINTEGER NEQN, IFLAG, NFE, KOP • INIT • JFLAG. KFLAGREAL Y(NEQN) ,T,POUT,RELERR,ABSERR,H,YP(NEQN),F1(NEQN),F2(NEQN),F3(NEQN).F4(NEQN).F5(NEQN),SAVRE,

2 SAV.EEXTERNAL FREAL A.AE.DT,EE.EEOET,ESTTOL, ET,HMIN,REMIN,RER,S.

SCALE.TOL.,TOLN,U26,EPSP1,EPS,YPKINTEGER K, MAXNPE, MFLAGREAL AMAX1,AMIN1DATA REMIN/1.E-12/DATA MAXNFE/3000/IF(NEQN.LT.1) GO TO 10IF((RELERRLTO.0) .OR. (ABSERR.LTO.0)) GO TO 10MFLAG=IABS(IFLAG)IF((MFLAG.EQ.0) 0R. (MFLAG.GT.8)) GO TO 10IF (MFLAG NE.1) GO TO 20EPS=1.0

5 EPS=EPS/2.0EPS1=EPS+1 0IF (EPSP1.GT.10) GO TO 5U26=26 OEPSGO TO 50

10 IFLAG=8RNTURN

20 IF ((T.EQTOUT).AND.(KFLAG.NE.3)) GO TO 10IF (MFLkG.NE.2) GO TO 25IF ((KFLAG.EQ.3).OR.(INITEQO)) GOPO 45IF (KFLAG.EQ.4) GO TO 40IF ((KFLAG.EQ.5) AND. (ABSERR.EQ.O.0)) GO TO 30IF ((KFLAG.EQ.6) AND. (RELERR.LE.SAVRE) AND.

(ABSERR LE SAVAE)) GO TO 30GO TO 50

25 IF (IFLAG.EQ.3) GO TO 45IF (IFLAG.EQ.4) GO TO 40IF ((IFLAG.EQ.5).AND.(ABSERILGT.0.0)) GO TO 45

30 STOP40 NFE=0

IF (MFLAG.EQ.2) GO TO 5045 IFLAG=JFLAG

IF (KFLAG .EQ. 3) MFLAG=IABS (IFLAG)50 JFLAG=IFLAG

KFLAG-OSAVRE-RELERRSAVAE-PBSERRRER 2.OEPS+REMIN


VI

IF (RELERR .GE. ERR) GO TO 55RELERR=RERIFLAG=3KFLAG=3REJRN

55 DT=TOUT-TIF (MFLAG.EQ. 1) GO TO 60IF (INIT.EQ.0) GO TO 65GO TO 80

60 INIT=0KOP=OATCALL F(A.Y.YP)NFE=1IF (T . NE. TOUT) GO TO 65IFLAG=2RETURN

65 INIT=1H=ABS (DT)TOLN=0DO 70 K=1,NEQN

TOL=RELERR* ABS (Y (K) ) +ABSERRIF (TOL.LE.0.) GO TO 70TOLN=TOL.YPK=ABS (YP (K))IF (YPK*H**5 .GT. TOL) H=(TOL/YPK)**0.2

70 CONTINUEIF (TOLN.LE.0.0) 14=0.0N=ANAX1(H,U26*AI4AX1(ABS(T) ABS(DT)))JFLG=ISIGN(2 • IFLAG)

80 H=SIGN(H,DT)IF (ABS(h).GE. 2.OABS(DT)) KOP=KOP^1IF(KOP.NE.100) GO TO 85KOP= 0IFLAG=7RETURN

85 IF (ABS(DT).GT. U26AB5(T)) GO TO 95DO 90 K=1,NEQN

90 Y(K)=y(K)^Dr*YP(K)A=TOUTCALL F(A.YYP)NFE=NFE+ 1GO TO 300

95 OUTPUT=.FALSE.SCALE=2 . 0/RELERRAE=SCALE*ABSEP.R

100 HFAILD=.FALSE.HNIN=U26*ARS (T)DT=TOUT-TIF (ABS(IYr).GE. 2.OABS(H)) GO TO 200IF (ABS(DT).GT. ABS(h)) GO TO 150OUTPUT= . TRUE.H=DTGO TO 200

150 H=0.5*DT200 IF (NFE.LE.MAXNFE) GO TO 220

IFLAG=4KFLAG=4RETURN

220 CALL FEL(F.NEQN.Y,T,H.YPF1,F2F3, F4.F5.F1)NFE=NFE+5EEOET=0.0DO 250 K=1,NEQN

ET-ABS(Y(K) )+ABS(F1(K))+AEIF (E'T.GT0.0) GO TO 240IFLAG=5


VII

RJRN

240 EE=ABS((_2090.0*YP(K)+(21970.0*F3(K)_15048.0*F4(K)))+

1 (22528.0*F2(K)_27360.0*F5(K)))250 EEOET=AMAX1 (EEOET, EE/ET)

ES'rTOL=ABS(H) *Q*SCALE/7524000IF (ESDTOL .LE. 1.0) GO TO 260HFAILD= .TRUE.OUTPtJT= . FALSE.S=0.1IF (ESTTOL.LT.59049.0) S=0.9/ESTrOL**0.2H=SHIF (ABS(H) .GT. HMIN) GO TO 200IFLAG=6KFLAG6RFURN

260 T=T+hDO 270 K=1,NEQN

270 Y(K)=F1(K)A=TCALL F(A,Y,YP)NFE=NFE+1S=5.0IF (ESTTOL .GT. 1.889568E-4) S=0.9/ESTrOL**0.2IF (HFAILD) S=ANIN1(S,1.0)H=SIGN(AMAX1(S*ABS(H) ,RMIN) ,H)IF (OUTPUT) GO TO 300IF (IFLAG.GT.0) GO TO 100IFLAG=-2

FURN300 T=TOUT

IFLAG-2RFURN

SUBROUTINE FEHL(F,NEQN,Y,T,H.YP,F1,F2,F3,F4,F5,S)INTEGER NEONREAL Y(NEQN).T,H,YP(NEQN),F1(NEQN).F2(NEQN),

1 F3 (NEON), F4 (NEQN) F5(NEQN) , S(NEQN)REAL CHINTEGER KCH=H/4.0DO 221 K=1,NEQN

221 F5(K)=Y(K)+CHYP(K)CALL F(T+CH,F5,F1)CH=3 .0*H/32 .0DO 222 K=1,NEQN

222 F5(K)=Y(K)+CH*(YP(K)+3.0*F1(K))CALL F(T+3.0h/8.0.F5,F2)CH=H/2197 .0DO 223 K=1,NEQN

223 F5(K)=Y(K)+CH*(1932.0*YP(K)+(7296.0*F2(K)_7200.0*F1(K)))CALL F(T+12.0*h/13.0,F5,F3)CH=H/4104 .0DO 224 K=1,NEQN

224 F5(K)=Y(K)+CH*((8341.0*YP(K)_845.0*F3(K))+

1 (29440.0*F2(K)_32832*F1(K)))CALL F(T+H,F5,F4)CH=H/20520.0DO 225 K=1,NEQN

225 F1(K)=Y(K)+CH*((_6080.O*YP(K)+(92950*F3(K)_

1 5643.0*F4(K)))+(41040.0*F1(K)_28352.0*F2(K)))CALL F(T+H/2.0,F1,F5)CH=H/7618050 .0000DO 230 K=1,NEQN

230 S(K)=Y(K)+CH*((902880.0*YP(K)+(3855735.0*F3(K)_

1 1371249.0*F4(K)))+(3953664.0*F2(K)+

2 277020.OF5(K)))


VIII

returnend

SUBROUTINE inter(a,b,c, ind,vol)real b(ind+l).c(ind+1)vol=0 .0b(1)=0.0c(1)=ado 1 i=1,ind-1if (c(i+l).lt.0.0) go to 99vol=vol+((c(i)+c(i+1)))*(b(i+l)**2_b(i)**2)*3.1415926continue

99 returnend

References

Andrews, E. H., Pingsheng, H. & Viachos, C. 1982 Adhesion of epoxy-resin to

glass. Proc. R. Soc. Lond., A381, 345-360.

Becker, W. M. 1986 The world of the cell. Benjamin/Cummings Publishing

Company Press.

Bentall, R. H. & Johnson, K. L. 1967. Slip in the rolling contact of two dissimilar

elastic rollers. mt. J. Mech. Sci., 9, 389 404.

Bliem, R. 1989 A need for systematic investigations into the material properties

of cultured animal cells. Trend in biotechnology, 7, 197-200.

Boussinesq, J. 1885 Application des Potentiels a I'Etude I'Equilibre et du Mou-

vement des Solides Elastiques. Gathier-Villars, Paris.

Bradley, R. S. 1932 The cohesive force between solid surfaces and the surface

energy of solids. Phil. Mag. , 13, 853-862.

Briggs, G. A. D. & Briscoe B. J. 1977 The effect of surface topography on the

adhesion of elastic solids. J. Phys. D. Appl. Phys., 10, 2453 2466.

Briscoe, B. J. 1987 Discrete particle-wall interactions in powder flow. Chem.

Eng. Sci., 42, 713 723.

Briscoe, B. J. & Panesar, S. 5. 1991 The application of the blister test to an

elastomeric adhesive. Proc. R. Soc. Lond., A433, 23 43.

162

REFERENCES 163

Briscoe, B. J. 1994 Interactions laws and the rheology of assemblies. AIChE 1st

mt Particle Technology Forum, Dever, USA.

Brown, D. J. 1993 Encapsulated W/O/W emulsions of ionic species; a study

to enable the preparation and examination of their release mechanisms.

MPhii/PhD Transfer Report, Imperial College, Univ. London.

Chang, K. S. & Olbricht, W. L. 1993 Experimental studies of the deformation of

a synthetic capsule in extensional flow. J. Fluid Mechanics, 250, 587 608.

Chaplain, M. J. S. 1990 & Sleeman, B. D. An application of membrane theory

to tip morphogenesis in Axetabularia. J. theor. Biol., 146, 177-200.

Chaplain, M. J. S. 1992 The strain energy function of an ideal plant cell wall.

J. theor. Biol., 163, 77 97.

Cole, K. S. 1932 Surface forces of the arbacia egg. J. Cell Comp. Physiol., 1,

1 9.

Darcy, H. 1856 Les Fontaines Publiques de la Viii de Dijon. Victor Dalmont,

Paris, 1856.

Daily B., E. L. Elson & C. I. Zahalak 1984 Cell poking: determination of the

elastic area compressiblity modulus of the erythrocyte membrane. Biophy.

J., 45, 671-682.

de Boer, J. II. 1936 The influence of van der Waals' forces and primary bonds

on binding energy, strengh and orientation with special reference to some

artificial resins. Trans Faraday Soc., 32, 10-38.

Derjaguin, B. V., Muller, V. M. & Toporov, Y. P. 1975 Effect of contact defor-

mations on the adhesion of particles. J. Coil. mt. Sci., 53, 314 326.

REFERENCES

164

Derjaguin, B. V., Krotova, N. A & Smilga, V. P. 1978 Adhesion of solids.

Consultants Bureau Press, New York.

Derjaguin, B. V., Muller, V. M. & Toporov, Y. P. 1980 On different approaches

to the contact mechanics. J. Coil. mt. Sci., 73, 293.

Ducker, W. A., Sender, T. J., & Pashley, R. M. 1991 Direct measurement of

colloidal forces using an atomic force microscope. Nature, 353, 239 241.

Evans I. D. & Lips A. 1990 Concentration dependence of the linear elastic

behaviour of model microgel dispersions. J. Cliem. Soc. Faraday Trans.,

86(20), 3413 3417.

Evans, E. A. & Skalak, R. 1979 Mechanics and Thermodynamics of Biomem-

branes. CRC Critical Reviews in Bioengineering. Vol 3, CRC Press, Boca

Raton.

Elson, E. L. 1988 Cellular mechanics as an indicator of cytoskeletal structure

and function. Annu. rev, of Biophy. and Biophysical Chem. , 17, 397 430

Feng, W. W., & Yang, W. H. 1973 On the contact problem of an inflated

spherical nonlinear membrane. J. Appi. Mech., ASME, 40, 209 214.

Feng, W. W. 1992 Viscoelastic behaviour of elastomeric membrane. J. Appi.

Mech., ASME, 59, 29 34.

Ferziger, J. H. 1981 Numerical methods for engineering application. Wiley, New

York.

Fisher, L. R. 1993 Forces between biological surface. Faraday Trans., J. Chem.

Soc., 89, 2567 2582.

REFERENCES

165

Fuller, K. N. C. & Tabor, D. 1975 The effect of surface roughness on the adhesion

of elastic solids. Proc. R. Soc. Lond., A345, 327-342.

Fung, Y. C. 1966 . Theoretical considerations of the elasticity of red cells and

small blood vessels. Fed. Proc. Symp, Microcirc., 25, 1761 1772.

Fung, Y. C. & Tong, P. 1968 Theory of the sphering of red blood cells. J.

Biophys., 8, 175-198.

Fung, Y. C. 1993 Biomechanics, 2nd. Springer-Verlag.

Fry, S. C., Smith, R. C., Renwick, K. F., Martin, D. J., Hodge, S. K. & Matthews

K. J. 1992 Xyloglucan endotransglycosylase, a new well-loosening enzyme

activity from plants. J. Biochem., 282, 821 828.

Gardner, W. R. & Ehlig, C. F. 1965 Physical aspects of the internal water

relations of plant leaves. Plant Phyiol., 40, 705 710.

Green, A. E., & Adkins, J. E. 1970 Large elastic deformation, 2nd. pp.168

Oxford University Press.

Greenwood, J. A. & Johnson, K. L. 1981 The mechanics of adhesion of vis-

coela.stic solids. Phil Mag., A43, 697 711.

Halpern, D., & Secomb, T. W. 1992 The squeezing of red blood cells through

parallel-sided channels with near-minimal widths. J. Fluid Mech., 244,

307 322.

Hamaker, H. C. 1937 The London-Van der Waals' attraction between spherical

particles. Physica, 4, 1058-1072.

Hertz, H. 1882 Uber die Berührung fester elatischer Körper. Zeitschrift für

Reine und Angewandte Mathematik., 92, 156 171.

REFERENCES

166

Hettiaratchi, D. R. P. & O'Callaghan J. R. 1974 A membrane model to plant

cell extension. J. theor. Biol., 45, 459-465.

Hettiaratchi, D. R. P. & O'Callaghan J. R. 1978 Structural mechanics of plant

cell. J. theor. Biol., 74, 235-257.

Johnson, R. M. 1987. Properties of red blood cells. In Handbook of Bioengineer-

ing. (ed. R. Skalak &' S. Chien), Chapter 12, Mcgraw-Hill, New York.

Israelachvili, J. N. & Tabor, D. 1972 The measurement of van der Waals dis-

persion forces in the range 1.5 to 130 nm. Proc. R. Soc. Lond., A331,

19 38.

Israelachvili, J. N. 1991 Intermolecular and surface forces, 2nd edition,. Aca-

demic Press INC.

Jalsenjak, I., Nicolaidou, C. T., & Nixon, J. R. 1976 The in vitro dissolution

of phenobarbitone sodium from ethyl cellulose microcapsules. J. Pharm.

Pharmac., 28, 912 914.

Jay, A. W. L., & Edwards, M. A. 1968 Can. J. Physiol. and Pharmacol., 46,

731-737.

Johnson, K. L. 1968. An experimental determination of the contact stresses be-

tween plastically deformed cylinders and spheres. In Engineering Plasticity.

(ed. J. Heyman & F. A. Leckie), pp. 179 183. Cambrige University Press.

Johnson, K. L., Kendall, K. & Roberts, A. D. 1971 Surface energy and the

contact of elastic solids. Proc. R. Soc. Lond., A324, 301 313.

Johnson, K. L., O'Connor, J. J. & Woodward, A. C. 1973 The effect of the

indenter elasticity on the Hertzian fracture of brittle materials. Proc. R.

Soc. Lond., A334, 95 117.

REFERENCES

167

Johnson, K. L. 1975. Non-Hertzian contact of elastic spheres. In The mechanics

of the contact between deformable bodies. (ed. A. D. de Pater & J. J. Kalker),

pp. 26-40. Deift University Press.

Johnson, K. L. 1985 Contact mechanics. Cambrj, e University Press.

Johnson, K. L. 1993 The mechanics of adhesion, deformation and contamination

in friction. Proc. 20th. Leeds-Lyon Symp. on Tribology, Lyon, 1993..

Kendall, K. 1988 Theoretical aspects of solid solid adhesion. Science

Progress, 72, 155-171.

Kendall, K. 1994 Cracks at adhesive interfaces. J. Adhesion Sci. Tech., 8,

1271 1284.

Lardner, T. J., & Pujara, P 1980 Compression of spherical cells. In Mechanics

Today, vol. 5. (ed. S. Nemat-Nasser), pp. 161-176. Pergamon.

Lardner, T. J., & Pujara, P. 1978 On the contact problem of a highly inflated

spherical nonlinear membrane. J. Appi. Mech., ASME, 45, 202 203.

Lifshitz, E. M. 1956 The theory of molecular attractive -c ' W between solids.

Soviet Phys. JETP (Engi. Transi.), 2, 73-83.

Loftsson, T., & Kristmundsóttir, T. 1993 Microcapsules containing water-soluble

cyclodextrin inclusion complexes of water-insoluble drugs. In Polymeric

delivery systems - properties and applications, (ed. M. A. El-Nokaly, D. M.

Piatt and B. A. Charpentier), pp. 168-189. ACS symposium series 520.

Luckham, P. F., & Costello B. A. Recent developments in the measurement of

interparticle forces. Adv. Colloid Jut. Sci., 44, 183-240.

REFERENCES

168

Maugis, D. 1985 Subcritical crack-growth, surface energy, fracture toughness,

stick-slip and embrittlement. J. Mater. Sci., 20, 3041-3073.

Maugis, D. 1992. Adhesion of spheres- the JKR-DMT transition using a Dugdale

model. J. Coil. mt. Sci., 150, 243 269.

Mooney, M. 1940 A theory of large elastic deformation. J. Appi. Phys., 11,

582 590.

Muller, V. M., Yushchenko, V. S. & Derjaguin, B. V. 1980 On the influence of

molecular force on the deformation of an elastic sphere and its sticking to

a rigid plane. J. Coil. mt. Sci., 77, 91 101.

Muller, V. M., Yushchenko, V. S. & Derjaguin, B. V. 1983 General theoretical

consideration of the influence of surface forces on contact deformations and

the reciprocal adhesion of elastic spherical particles. J. Coil. mt. Sci., 92,

92 101.

Nikias, K. J. 1992 Plant biomechanics. The University of Chicago Press.

Ogden, R. W. 1972 Large deformation isotropic elasticity - on the correlation of

theory and experiment for incompressible rubber-like solids. Proc. R. Soc.

Lond., A326, 565 584.

Petersen N. 0., W. B. McConnaughey & E. L. Elson 1981 Investigations of

structural determinants of cell shape. Comments Mol. Cell. Biophys., 1,

135 147.

Prapaitrakul, W., & Whitworth, C. W. 1990 Effect of excipients and pressure

on physical properties. Drug Dev. md. Pharm., 16, 1427-1434.

Rivlin, R. S. 1948 Large elastic deformation of isotropic materials IV. Further

developments of the general theory. Phil. Trans. Roy. Soc., A241, 379 397.

REFERENCES

169

Sebastian K. S. 1994 Indentation hardness of polymers. PhD thesis, Univ.

London.

Skalak R., Tozeren A., Zarda R. P. & Chien S. 1973 Strain energy function of

red blood cell membranes. Biophs. J., 13, 245 264.

Shipway P. H. & I. M. Hutchings 1993 Attrition of brittle spheres by fracture

under compression and impact load. Powder Tech., 76, 23-30.

Taber, L. A. 1982 Large deflection of a fluid-filled spherical shell under a point

load. J. Appi. Mech., ASME, 49, 121-128.

Tabor, D. & Winterton R. H. S. 1969 The direct measurement of normal and

retarded van der Waals forces. Proc. R. Soc. Lond., A312, 435 450.

Tabor, D. 1977 Surface force and surface interactions. J. Coil. mt. Sci., 58,

2 13.

Tabor, D. 1980 Role of molecular forces in the contact deformations. J. Coil.

mt. Sci., 73, 294.

Tabor, D. 1987 Adhesion of solids. In Tribology in particulate technology. (ed.

B. J. Briscoe & M. J. Adams), pp. 206-219. lOP Publishing Ltd press.

Tatara, Y. 1989 Extensive theory of force-approach relations of elastic spheres in

compression and in compact. J. Eng. Mater. Tech., ASME, 111, 163-168.

Tatara, Y. 1991 On compression of rubber elastic sphere over a large range

displacements-partl:theoretical study. J. Eng. Mater. Tech., ASME, 113,

285-291.

REFERENCES

170

Tatara, Y., Shima, S. & Lucero, J. C. 1991 On compression of rubber elastic

sphere over a large range displacements-part2:comparison of theory and

experiment. J. Eng. Mater. Tech., ASME, 113, 292-295.

Timoshenko, S. P. Coodier, J. N 1970 Theory of elasticity. 3rd Ed., McGraw-

Hill Ltd., pp. 398-422.

Treloar, L. R. G 1970 The physics of rubber elasticity. 3rd Ed., Oxford Univ.

Press, pp. 64-98.

Williams, D. R., Liu, K. K. &? Briscoe, B. J. 1995. A novel experimental method

for characterising the mechanical properties of single microscopic particles.

To be published in Rev. Sci. Instrum..

Wu, H., Spence, R. D. & Sharpe, P. J. H. 1988 Plant cell wall elasticity II.

Polymer elastic properties of the microfibrils. J. theor. Biol., 133, 239 253.

Wu, S. 1982. Polymer interface and adhesion. Marcel Dekker Inc press.

Yoffe, E. H. 1984. Modified Hertz theory for spherical indentation. Phil Mag.,

A50, 815-828.

Yoneda, M. 1973 Tension at the surface of sea urchin eggs on the basis of

liquid-drop concept. Advanc. in Biophy., 4, 153 190.

Zahalak G. I., W. B. McConnaughey & E. L. Elson 1990 Determination of cellu-

lar mechanical properties by cell poking, with an application to leukocytes.

J. Biomech. Eng., ASME, 112, 283-294.

Zhang, Z., Saunders, R., & Thomas, C. R. 1994 Micromanipulation measure-

ments of the bursting strength of single microcapsules. IChem Even., 2,

722-724.

REFERENCES

171

Zhang Z., M. A. Ferenczi & C. R. Thomas 1992 A micromanipulation technique with

a theoretical cell model for determining mechanical properties of single mammalian

cells. Chem. Eng. Scie., 47, 1347-1354.

Additional References

Kendall K. & T. P. Weihs 1992 Adhesion of nanoparticles within spray-dried

agglomerates. .1. Phys. D. App!. Phys., 25, A3-A8.

Krishnan S., A. A. Busnaina, D. S. Rimai and L. P. DeMejo 1994 The adhesion-

induced deformation and the removal of submicrometer particles. J. Adhes. Sci. &

Tech., 8, 1357-1370.

Rimai D. S., L. P. Demejo and R. C. Bowen 1994 Mechanics of particle adhesion. .1.

Adhes. Sci. & Tech., 8, 1333-1355.

Shipway P. H. and I. M. Hutchings 1993 Fracture of brittle spheres under compression

and impact loading: I -Elastic stress distributions, Phil. Mag. A 67, 1389-1404.

Shipway P. H. and I. M. Hutchings 1993 Fracture of brittle spheres under compression

and impact loading: II -Results for lead-glass and sapphire spheres, Phi!. Mag. A 67,

1405-1421.

)

Documents

Department of Chemical Engineering, June 20, 1995