Table Of ContentEngineering Materials and Processes
Series Editor
Professor Brian Derby, Professor of Materials Science
Manchester Materials Science Centre, Grosvenor Street, Manchester, M1 7HS, UK
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Sensors and Spintronics Based on Ferroelectrics
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William W. Sampson
Modelling Stochastic Fibrous
®
Materials with Mathematica
123
William W. Sampson, PhD
School of Materials
University of Manchester
Sackville Street
Manchester
M60 1QD
UK
ISBN 978-1-84800-990-5 e-ISBN 978-1-84800-991-2
DOI 10.1007/978-1-84800-991-2
Engineering Materials and Processes ISSN 1619-0181
A catalogue record for this book is available from the British Library
Library of Congress Control Number: 2008934906
© 2009 Springer-Verlag London Limited
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Preface
This is a book with three functions. Primarily, it serves as a treatise on the
structure of stochastic fibrous materials with an emphasis on understanding
howthepropertiesoffibresinfluencethoseofthematerial.Forsomeofus,the
structureoffibrousmaterialsisatopicofinterestinitsownright,andweshall
seethattherearemanyfeaturesofthestructurethatcanbecharacterisedby
ratherelegantmathematicaltreatments.Theinterestofmostresearchershow-
ever is the manner in which the structure of fibrous networks influences their
performanceinsomeapplication, suchas theability of a non-woven textile to
captureparticlesinafiltrationprocessorthepropensityofcellstoproliferate
onanelectrospunfibrousscaffoldintissueengineering.Thesecondfunctionof
thisbookisthereforetoprovideafamilyofmathematicaltechniquesformod-
elling, allowing us to make statements about how different variables influence
the structure and properties of materials. The final intended function of this
bookistodemonstratehowthesoftwareMathematica1canbeusedtosupport
themodellingwork,makingthetechniquesaccessibletonon-mathematicians.
We proceed by assuming no prior knowledge of any of the main aspects
of the approach. Specifically, the text is designed to be accessible to any sci-
entist or engineer whatever their experience of stochastic fibrous materials,
mathematical modelling or Mathematica. We begin with an introduction to
each of these three topics, starting with defining clearly the criteria that clas-
sify stochastic fibrous materials. Weproceedto consider the reasons for using
modelstoguideourunderstandingandspecificallywhyMathematicahasbeen
chosen as a computational aid to the process.
Importantly,thisbookisintendednotjusttoberead,buttobeused.Ithas
been written in a style intended to allow the reader to extract all the impor-
tant relationships from the models presented without using the Mathematica
1 A trial version of Mathematica 6.0 can be downloaded from
http://www.wolfram.com/books/resources, by entering the licence number
L3250-9882. Mathematica notebook files containing the code presented in each
chapter can be downloaded from http://www.springer.com/978-1-84800-990-5.
vi Preface
examplesprovided.However,foracomprehensiveunderstandingofthescience
underlyingthemodels,readerswillbenefitfromrunningthecodeandediting
ittoprobefurtherthedependenciesweidentifyanddiscuss.TheMathematica
code embedded in the text includes numbers added by Mathematica on eval-
uation. Where a line of code begins with ‘In[1]:=’ this indicates that a new
session on the Mathematica kernel has been started and previously defined
variables, etc. no longer exist in the system memory.
When preparing the manuscript, some consideration was given as to the
appropriate amount of Mathematica code to include. In common with many
scientists, the author’s first approach to any theoretical analysis involves the
traditionaltoolsofpaperandpencil,withtheuseofMathematicabeingintro-
duced after the initial formulation of the problem of interest. Typically how-
ever,onceprogresshasbeenmadeonaproblem,theearlymanipulations,etc.
are subsequently entered into Mathematica so that the full treatment is con-
tainedinasinglefile.Asarule,weseektoreplicatethisapproachbyproviding
many of the preliminary relationships and straightforward manipulations as
ordinary typeset equations before turning to Mathematica for the more de-
mandingaspectsoftheanalysis.Occasionally,wheretheoutputsofrelatively
simple manipulations are required for subsequent treatments, Mathematica is
invoked at an earlier stage.
Consideration was given also as to whether to include Mathematica code
forthegenerationofplots,ortoprovidetheseonlyintheformoffigureswith
supplementarydetail,suchasarrows,etc.Oneofthegreatadvantagesofwork-
ing with Mathematica is that its advanced graphics capabilities allow rapid
generationofplotsandsurfacesrepresentingfunctions;surfacescanberotated
using a mouse or other input device and dynamic plots with interactivity are
readily generated. When developing theory, the ability to visualise functions
guidestheprocessandcanprovidevaluablereassurancethatfunctionsbehave
inawaythatisrepresentativeofthephysicalsystemofinterest.Accordingly,
the Mathematica code used to generate graphics is provided in the majority
of instances and graphics are not associated with figure numbers, but instead
are shown as the output of a Mathematica evaluation. Where graphics are
associated with a figure number, the content is either a drawing to guide our
analysisoracollectionofresultsfromseveralMathematicaevaluations;inthe
lattercase,plotsorsurfaceshavebeentypicallygeneratedusingMathematica
and exported to graphics software for the addition of supplementary detail
and annotation.
The manuscript has benefited greatly from the comments of Kit Dodson,
NicolaDooley,RaminFarnoodandSteveI’Anson,whoprovidedhelpfulfeed-
backonanearlydraft.Iwouldliketothankeachofthemforbeingsogenerous
of their time and for being so diligent in their attention to detail. Kit Dodson
deserves particular thanks and recognition for introducing me to statistical
geometry, stochastic modelling and Mathematica when I spent time with his
research group at the University of Toronto in the early 1990s. Many of the
Preface vii
outcomesfromourfifteenyearsoffruitfulandenjoyableresearchcollaboration
are included in this monograph.
Thanks are also due to Maryka Baraka of Wolfram Research for support
and advice, to Steve Eichhorn for permission to reproduce the micrograph
of an electrospun nanofibrous network in Figure 1.1 and to Steve Keller for
permission to reproduce Figures 5.7 and 5.8. I would like to thank Taylor
and Francis Ltd. for permission to reproduce Figure 3.7, Journal of Pulp and
Paper Science for permission to reproduce Figure 6.1 and Wiley-VCH Verlag
GmbH & Co. KGaA for permission to reproduce Figure 7.1.
Manchester Bill Sampson
February, 2008
Contents
1 Introduction............................................... 1
1.1 Random, Near-Random and Stochastic..................... 3
1.2 Reasons for Theoretical Analysis .......................... 9
1.3 Modelling with Mathematica .............................. 11
2 Statistical Tools and Terminology.......................... 15
2.1 Introduction ............................................ 15
2.2 Discrete and Continuous Random Variables................. 15
2.2.1 Characterising Statistics............................ 16
2.3 Common Probability Functions ........................... 29
2.3.1 Bernoulli Distribution.............................. 29
2.3.2 Binomial Distribution.............................. 31
2.3.3 Poisson Distribution ............................... 35
2.4 Common Probability Density Functions .................... 37
2.4.1 Uniform Distribution .............................. 38
2.4.2 Normal Distribution ............................... 39
2.4.3 Lognormal Distribution ............................ 42
2.4.4 Exponential distribution ........................... 45
2.4.5 Gamma Distribution............................... 46
2.5 Multivariate Distributions ................................ 49
2.5.1 Bivariate Normal Distribution ...................... 51
3 Planar Poisson Point and Line Processes .................. 55
3.1 Introduction ............................................ 55
3.2 Point Poisson Processes .................................. 55
3.2.1 Clustering ........................................ 56
3.2.2 Separation of Pairs of Points........................ 63
3.3 Poisson Line Processes ................................... 71
3.3.1 Process Intensity .................................. 73
3.3.2 Inter-crossing Distances ............................ 81
x Contents
3.3.3 Statistics of Polygons .............................. 83
3.3.4 Intrinsic Correlation ............................... 94
4 Poisson Fibre Processes I: Fibre Phase ....................105
4.1 Introduction ............................................105
4.2 Planar Fibre Networks ...................................105
4.2.1 Probability of Crossing.............................110
4.2.2 Fractional Contact Area............................115
4.2.3 Fractional Between-zones Variance...................117
4.3 Layered Fibre Networks ..................................132
4.3.1 Fractional Contact Area............................132
4.3.2 In-plane Distribution of Fractional Contact Area ......137
4.3.3 Intensity of Contacts...............................146
4.3.4 Absolute Contact States............................150
5 Poisson Fibre Processes II: Void Phase ....................159
5.1 Introduction ............................................159
5.2 In-plane Pore Dimensions.................................160
5.3 Out-of-plane Pore Dimensions.............................171
5.4 Porous Anisotropy.......................................174
5.5 Tortuosity ..............................................182
5.6 Distribution of Porosity ..................................184
5.6.1 Bivariate Normal Distribution ......................185
5.6.2 Implications for Network Permeability ...............191
6 Stochastic Departures from Randomness ..................195
6.1 Introduction ............................................195
6.2 Fibre Orientation Distributions............................196
6.2.1 One-parameter Cosine Distribution ..................196
6.2.2 von Mises Distribution .............................199
6.2.3 Wrapped Cauchy Distribution ......................201
6.2.4 Comparing Orientation Distribution Functions ........202
6.2.5 Fibre Crossings ...................................206
6.2.6 Crossing Area Distribution .........................211
6.2.7 Mass Distribution .................................216
6.3 Fibre Clumping and Dispersion ...........................217
6.3.1 Influence on Network Parameters....................222
7 Three-dimensional Networks...............................241
7.1 Introduction ............................................241
7.2 Network Density ........................................244
7.2.1 Crowding Number.................................246
7.3 Intensity of Contacts.....................................248
7.4 Variance of Porosity .....................................250
Contents xi
7.5 Variance of Areal Density ................................253
7.6 Sphere Caging ..........................................258
References.....................................................265
Index..........................................................275
