Diffusion in Solids: Fundamentals, Methods, Materials, Diffusion-Controlled Processes

644 Pages·2007·13.939 MB·English

by  Professor Dr. Helmut Mehrer (auth.)

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Helmut Mehrer
Diffusion in Solids
Fundamentals,Methods,Materials,
Diffusion-ControlledProcesses
WithFiguresandTables
123
ProfessorDr.HelmutMehrer
Universita¨tMu¨nster
Institutfu¨rMaterialphysik
Wilhelm-Klemm-Str.
Mu¨nster
Germany
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Max-Planck-Institutfu¨rFestko¨rperforschung,Heisenbergstrasse,Stuttgart,Germany
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Preface
Diffusion is the transport of matter from one point to another by thermal
motionofatomsormolecules.Itisrelativelyfastingases,slowinliquids,and
very slowin solids.Diffusion playsa key roˆle in many processesasdiverse as
intermixing of gases and liquids, permeation of atoms or molecules through
membranes,evaporationofliquids,dryingoftimber,dopingsiliconwafersto
make semiconductor devices, and transport of thermal neutrons in nuclear
power reactors. Rates of important chemical reactions are limited by how
fast diffusion can bring reactants together or deliver them to reaction sites
on enzymes or other catalysts.
Diffusioninsolidmaterialsis the subjectofthis book.Alreadyinancient
times reactions in the solid state such as surface hardening of steels were
in use, which according to our present knowledge involves the diffusion of
carbon atoms in the crystallattice of iron. Nevertheless,until the end of the
nineteenth century the paradigm‘Corpora non agunt nisi fluida’ was widely
accepted by the scientific community. It was mainly due to the pioneering
work of William Roberts-Austen and Georg von Hevesy that this paradigm
had to be abandoned.
Diffusion in solids is fundamental in the art and science of materials and
thus an important topic of solid-state physics, physical chemistry, physical
metallurgy, and materials science. Diffusion processes are relevant for the
kineticsofmanymicrostructuralchangesthatoccurduringpreparation,pro-
cessing, and heat treatment of materials.Typical examples are nucleation of
new phases, diffusive phase transformations,precipitation and dissolution of
asecondphase,homogenisationofalloys,recrystallisation,high-temperature
creep, and thermal oxidation. Diffusion and electrical conduction in ionic
conductors are closely related phenomena. Direct technological applications
of diffusion concern, e.g., doping during fabrication of microelectronic de-
vices, solid electrolytes for batteries and fuel cells, surface hardening of steel
through carburisation or nitridation, diffusion bonding, and sintering.
Appreciable diffusion in solids mostly takes place at temperatures well
aboveroomtemperature.Knowledgeofdiffusionisthereforeparticularlyim-
portantfor scientists who designmaterialsfor elevated temperaturesand for
engineerswhobuildequipmentforoperationatsuchtemperatures.However,
processes connected with diffusion at room temperature pose problems, too.
VIII
Creep, atmospheric corrosion, and embrittlement of solders are among the
moreprominentofthose.Withthedownscalingofmicroelectroniccircuitsto
nanometer dimensions, diffusion and electromigration in these circuits must
be taken into account.
A deeper knowledge about diffusion requires information on the position
ofatomsandhowtheymoveinsolids.Theatomicmechanismsofdiffusionin
crystallinesolidsarecloselyconnectedwithdefects.Pointdefectssuchasva-
cancies orinterstitials arethe simplestdefects andoftenmediate diffusionin
crystals. Dislocations, grain-boundaries, phase boundaries, and free surfaces
are other types of defects. They can act as high-diffusivity paths (diffusion
short circuits), because the mobility of atoms along such defects is usually
much higher than in the lattice. In solids with structural disorder such as
glasses or crystals with highly disordered sublattices the concept of defects
is no longer useful. Nevertheless, diffusion is fundamental for transport of
matter and for ionic conduction in disordered materials.
The contentofthis bookis dividedinto sevenparts.After a historicalin-
troduction and a diffusion bibliography, Part I introduces basic concepts of
diffusioninsolidmatter suchascontinuumdescription,randomwalktheory,
pointdefects,atomicmechanisms,correlationeffects,dependenceofdiffusion
ontemperature,pressureandisotopemass,diffusionwithdrivingforces,and
some remarks about the relation between diffusion and thermodynamics of
irreversible processes. The necessary background is a course in solid-state
physics. In Part II we describe experimental methods for the determina-
tion of diffusion coefficients in solid matter. Direct methods based on Fick’s
laws and indirect methods such as anelastic relaxation, internal friction, nu-
clearmagneticrelaxation,M¨ossbauerspectroscopy,quasielasticneutronscat-
tering, impedance spectroscopy, and spreading resistance measurements are
treated. In further parts we provide access to information on diffusion in
various types of materials such as metals, intermetallics and quasicrystalline
alloys(PartIII),semiconductors(PartIV),ionicmaterialsincludingfastion
conductors(Part V),metallicandoxideglasses(Part VI).Finally,rapiddif-
fusion paths suchas grain-boundarydiffusion anddiffusion in nanomaterials
are considered (Part VII). Although these parts cannot replace a compre-
hensivedatacollection,typicalup-to-dateresourcesavailableondiffusionfor
various types of materials are noted.
A thorough understanding of diffusion in materials is crucial for materi-
als development and engineering. Graduate students in solid state physics,
physicalmetallurgy,physicalandinorganicchemistry,andgeophysicalmate-
rialswillbenefitfromthisbookaswillphysicists,chemistsandmetallurgists,
working in academia and industry.
Mu¨nster, May 2007 Helmut Mehrer
Acknowledgements
Likeanyauthorofascientificbook,Iamindebtedtopreviouswritersondif-
fusion and allied subjects. In particular I acknowledge the encouragementof
colleaguesandfriendswhoprovidedinvaluableassistanceduringtheprepara-
tionofthe book.Prof.HartmutBracht,Prof.KlausFunke,andDr.Nikolaas
Stolwijk, all from the University of Mu¨nster, have read many chapters as
they unfolded. Prof.GaborErdelyi,Debrecen,Hungary,Prof.Andry Gusak,
Cherkassy, Ukraine, Dr. J.N. Mundy, USA, and Prof. Malcom Ingram, Uni-
versity of Aberdeen, Great Britain, stayed in my University as guest sci-
entist for some time and have also reviewed several chapters of the book.
Dr. M. Hirscher, Max-Planck-Institut fu¨r Metallforschung, Stuttgart, pro-
vided comments for the chapter on hydrogen diffusion. Prof. Gerhard Wilde
from the University of Mu¨nster made useful suggestions for the chapter on
nanomaterials.Prof.GraemeMurch,Newcastle,Australia,didagreatjobin
reading the whole book, polishing my English, and providing many helpful
suggestions.
To my collaborators Dr. Serguei Divinski, Dr. Arpad Imre, Dr. Halgard
Staesche and my PhD students and postdocs Dr. Robert Galler, Dr. Marcel
Salamon, Dr. Serguei Peteline, Dr. Eugene Tanguep Nijokep, Dr. Stephan
Voss, and to my secretary Sylvia Gurnik I owe many thanks for critically
readingpartsofthebook.Dr.ArpadImrewasagreathelpinpreparingmost
figures.Thecontributionsandconstructivecriticismsofallthesepersonswere
most helpful.
Contents
1 History and Bibliography of Diffusion .................... 1
1.1 Pioneers and Landmarks of Diffusion...................... 2
References ................................................. 16
1.2 Bibliography of Solid-State Diffusion...................... 18
Part I Fundamentals of Diffusion
2 Continuum Theory of Diffusion........................... 27
2.1 Fick’s Laws in Isotropic Media ........................... 27
2.1.1 Fick’s First Law.................................. 28
2.1.2 Equation of Continuity............................ 29
2.1.3 Fick’s Second Law – the ‘Diffusion Equation’ ........ 30
2.2 Diffusion Equation in Various Coordinates................. 31
2.3 Fick’s Laws in Anisotropic Media......................... 33
References ................................................. 35
3 Solutions of the Diffusion Equation ....................... 37
3.1 Steady-State Diffusion .................................. 37
3.2 Non-Steady-State Diffusion in one Dimension .............. 39
3.2.1 Thin-Film Solution ............................... 39
3.2.2 Extended Initial Distribution
and Constant Surface Concentration................ 41
3.2.3 Method of Laplace Transformation ................. 45
3.2.4 Diffusion in a Plane Sheet – Separation of Variables .. 47
3.2.5 Radial Diffusion in a Cylinder...................... 50
3.2.6 Radial Diffusion in a Sphere ....................... 51
3.3 Point Source in one, two, and three Dimensions ............ 52
References ................................................. 53
4 Random Walk Theory and Atomic Jump Process......... 55
4.1 Random Walk and Diffusion ............................. 56
4.1.1 A Simplified Model ............................... 56
4.1.2 Einstein-Smoluchowski Relation.................... 58
4.1.3 Random Walk on a Lattice ........................ 60
XII Contents
4.1.4 CorrelationFactor................................ 62
4.2 Atomic Jump Process................................... 64
References ................................................. 66
5 Point Defects in Crystals ................................. 69
5.1 Pure Metals ........................................... 70
5.1.1 Vacancies ....................................... 70
5.1.2 Divacancies...................................... 72
5.1.3 Determination of Vacancy Properties ............... 74
5.1.4 Self-Interstitials .................................. 79
5.2 Substitutional Binary Alloys ............................. 80
5.2.1 Vacancies in Dilute Alloys ......................... 81
5.2.2 Vacancies in Concentrated Alloys................... 82
5.3 Ionic Compounds....................................... 83
5.3.1 Frenkel Disorder ................................. 84
5.3.2 Schottky Disorder ................................ 85
5.4 Intermetallics .......................................... 86
5.5 Semiconductors ........................................ 88
References ................................................. 91
6 Diffusion Mechanisms .................................... 95
6.1 Interstitial Mechanism .................................. 95
6.2 Collective Mechanisms .................................. 97
6.3 Vacancy Mechanism .................................... 98
6.4 Divacancy Mechanism................................... 100
6.5 Interstitialcy Mechanism ................................ 100
6.6 Interstitial-substitutional Exchange Mechanisms............ 102
References ................................................. 103
7 Correlation in Solid-State Diffusion....................... 105
7.1 Interstitial Mechanism .................................. 107
7.2 Interstitialcy Mechanism ................................ 107
7.3 Vacancy Mechanism of Self-diffusion ...................... 108
7.3.1 A ‘Rule of Thumb’ ............................... 108
7.3.2 Vacancy-tracer Encounters ........................ 109
7.3.3 Spatial and Temporal Correlation .................. 112
7.3.4 Calculation of CorrelationFactors .................. 112
7.4 Correlation Factors of Self-diffusion....................... 115
7.5 Vacancy-mediated Solute Diffusion ....................... 116
7.5.1 Face-Centered Cubic Solvents...................... 117
7.5.2 Body-Centered Cubic Solvents ..................... 120
7.5.3 Diamond Structure Solvents ....................... 121
7.6 Concluding Remarks.................................... 122
References ................................................. 124