1) There are two approaches to describing grain boundaries mechanically: a continuous approach using dislocation density tensors and a discrete approach considering arrays of dislocations.
2) The continuous Frank-Bilby equation relates the Burgers vector density necessary to achieve compatibility at the interface between crystals to the affine transformations relating their lattices.
3) The discrete Read-Shockley model links the misorientation angle and spacing of edge dislocations forming low-angle symmetrical tilt grain boundaries. It predicts the intergranular energy as a function of angle.
Introduction to casting, Major classifications of casting, Casting terminology, Characteristics of molding sand, Constituents of foundry sand, Patterns and their types, Cores and types of cores, Gating system, Types of gates, Solidification, Riser system, Types of riser, Types of allowances, Directional Solidification, Defects in casting, Riser design(Chvorinov's rules), Advanced casting techniques:Shell molding, Permanent mould casting, Vacuum die casting, Low pressure die casting, Continuous casting, Squeeze casting, Slush casting, Vacuum casting, Die Casting, Centrifugal casting, Investment casting
This lecture describes the impact extrusion processes as well as the forces and deformations acting on the tools and work-piece in order to give insight into the relation between part design, process and tooling. Basic knowledge about the formability of metals and background in mechanical engineering is assumed.
In view of Fatigue & Fracture Mechanics, the slides give a insight off the effects of the Creep and Fatigue Crack growth.
As both the fatigue and creep phenomenon is dynamic in nature due to the involvement of parameter w.r.t time.
Thereby the J contour Integral parameter is replaced by C-Integral by adopting Hoff's Analogy and Power's Law.
Introduction to casting, Major classifications of casting, Casting terminology, Characteristics of molding sand, Constituents of foundry sand, Patterns and their types, Cores and types of cores, Gating system, Types of gates, Solidification, Riser system, Types of riser, Types of allowances, Directional Solidification, Defects in casting, Riser design(Chvorinov's rules), Advanced casting techniques:Shell molding, Permanent mould casting, Vacuum die casting, Low pressure die casting, Continuous casting, Squeeze casting, Slush casting, Vacuum casting, Die Casting, Centrifugal casting, Investment casting
This lecture describes the impact extrusion processes as well as the forces and deformations acting on the tools and work-piece in order to give insight into the relation between part design, process and tooling. Basic knowledge about the formability of metals and background in mechanical engineering is assumed.
In view of Fatigue & Fracture Mechanics, the slides give a insight off the effects of the Creep and Fatigue Crack growth.
As both the fatigue and creep phenomenon is dynamic in nature due to the involvement of parameter w.r.t time.
Thereby the J contour Integral parameter is replaced by C-Integral by adopting Hoff's Analogy and Power's Law.
The failure or fracture of a product or component as a result of a single event is known as mechanical overload. It is a common failure mode, and may be contrasted with fatigue, creep, rupture, or stress relaxation.
BRITTLE FAILURE
In a brittle overload failure, separation of the two halves isn’t quite instantaneous, but proceeds at a tremendous rate, nearly at the speed of sound in the material. The crack begins at the point of maximum stress, then grows across by cleavage of the individual material grains. One of the results of this is that the direction of the fracture path is frequently indicated by chevron marks that point toward the origin of the failure.
International Journal of Engineering Research and Development (IJERD)IJERD Editor
journal publishing, how to publish research paper, Call For research paper, international journal, publishing a paper, IJERD, journal of science and technology, how to get a research paper published, publishing a paper, publishing of journal, publishing of research paper, reserach and review articles, IJERD Journal, How to publish your research paper, publish research paper, open access engineering journal, Engineering journal, Mathemetics journal, Physics journal, Chemistry journal, Computer Engineering, Computer Science journal, how to submit your paper, peer reviw journal, indexed journal, reserach and review articles, engineering journal, www.ijerd.com, research journals,
yahoo journals, bing journals, International Journal of Engineering Research and Development, google journals, hard copy of journal
The failure or fracture of a product or component as a result of a single event is known as mechanical overload. It is a common failure mode, and may be contrasted with fatigue, creep, rupture, or stress relaxation.
BRITTLE FAILURE
In a brittle overload failure, separation of the two halves isn’t quite instantaneous, but proceeds at a tremendous rate, nearly at the speed of sound in the material. The crack begins at the point of maximum stress, then grows across by cleavage of the individual material grains. One of the results of this is that the direction of the fracture path is frequently indicated by chevron marks that point toward the origin of the failure.
International Journal of Engineering Research and Development (IJERD)IJERD Editor
journal publishing, how to publish research paper, Call For research paper, international journal, publishing a paper, IJERD, journal of science and technology, how to get a research paper published, publishing a paper, publishing of journal, publishing of research paper, reserach and review articles, IJERD Journal, How to publish your research paper, publish research paper, open access engineering journal, Engineering journal, Mathemetics journal, Physics journal, Chemistry journal, Computer Engineering, Computer Science journal, how to submit your paper, peer reviw journal, indexed journal, reserach and review articles, engineering journal, www.ijerd.com, research journals,
yahoo journals, bing journals, International Journal of Engineering Research and Development, google journals, hard copy of journal
Periodic material-based vibration isolation for satellitesIJERA Editor
The vibration environment of a satellite is very severe during launch. Isolating the satellitevibrations during
launch will significantly enhance reliability and lifespan, and reduce the weight of satellite structure and
manufacturing cost. Guided by the recent advances in solid-state physics research, a new type of satellite
vibration isolator is proposed by usingperiodic material that is hence called periodic isolator. The periodic
isolator possesses a unique dynamic property, i.e., frequency band gaps. External vibrations with frequencies
falling in the frequency band gaps of the periodic isolator are to be isolated. Using the elastodynamics and the
Bloch-Floquet theorem, the frequency band gaps of periodic isolators are determined. A parametric study is
conducted to provide guidelines for the design of periodic isolators. Based on these analytical results, a finite
element model of a micro-satellite with a set of designed periodic isolators is built to show the feasibility of
vibration isolation. The periodic isolator is found to be a multi-directional isolator that provides vibration
isolation in the three directions.
Periodic material-based vibration isolation for satellitesIJERA Editor
The vibration environment of a satellite is very severe during launch. Isolating the satellitevibrations during
launch will significantly enhance reliability and lifespan, and reduce the weight of satellite structure and
manufacturing cost. Guided by the recent advances in solid-state physics research, a new type of satellite
vibration isolator is proposed by usingperiodic material that is hence called periodic isolator. The periodic
isolator possesses a unique dynamic property, i.e., frequency band gaps. External vibrations with frequencies
falling in the frequency band gaps of the periodic isolator are to be isolated. Using the elastodynamics and the
Bloch-Floquet theorem, the frequency band gaps of periodic isolators are determined. A parametric study is
conducted to provide guidelines for the design of periodic isolators. Based on these analytical results, a finite
element model of a micro-satellite with a set of designed periodic isolators is built to show the feasibility of
vibration isolation. The periodic isolator is found to be a multi-directional isolator that provides vibration
isolation in the three directions.
X-Ray Line Profile Analysis of Chemically Deposited Nanostructured PBS FilmsIOSR Journals
Nanocrystalline films of PbS have been deposited on glass sustrates at room temperature by CBD
method. The structural parameters of PbS films have been studied by X-ray line profile analysis using
Williamson Hall and Modified Williamson Hall method. The crystallite sizes are found in between 4.99-53.9 nm,
strain in the films in the range of 7.4x10-4 – 2.82 x 10-3 and dislocation densities are found to be very high ~
1015-1016 m-
IOSR Journal of Mathematics(IOSR-JM) is an open access international journal that provides rapid publication (within a month) of articles in all areas of mathemetics and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications in mathematics. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
Research on Contact Characteristics between Bump End Effector and WaferIJRES Journal
In the IC industry, commonly used methods are wafer clamping friction transmission type and vacuum suction. Combining science and theological contact theory,the contact friction transmission characteristics when using the bump and transmission actuator wafer, the wafer and the end actuators. Starting from the material properties of the wafer by ANSYS simulation analysis in contact with the wafer bump deformation due to its own gravity, and verify that it meets the requirements of small deformation wafer transfer. Compute and solve the friction contact with the wafer bump bristles between.
IOSR Journal of Applied Physics (IOSR-JAP) is an open access international journal that provides rapid publication (within a month) of articles in all areas of physics and its applications. The journal welcomes publications of high quality papers on theoretical developments and practical applications in applied physics. Original research papers, state-of-the-art reviews, and high quality technical notes are invited for publications.
The ideal, perfectly regular crystal structures in which atoms are arranged in a regular way does not exist in actual situations. In actual cases, the regular arrangements of atoms disrupted . These disruptions are known as Crystal imperfections or crystal defects
The unexpected fractal signatures in fibonacci chains 10.06.16 - final - ff Klee Irwin
Quasicrystals are fractal because they are scale invariant and self similar. In this paper, a new cycloidal fractal signature possessing the cardioid shape in the Mandelbrot set is presented in the Fourier space of a Fibonacci chain with two lengths, L and S, where L/S = {\phi}. The corresponding pointwise dimension is 0.7. Various variations such as truncation from the head or tail, scrambling the orders of the sequence, changeing the ratio of the L and S, are done on the Fibonacci chain. The resulting patterns in the Fourier space show that that the fractal signature is very sensitive to the change in the Fibonacci order but not to the L/S ratio.
Dynamics of Twointeracting Electronsinthree-Dimensional LatticeIOSR Journals
The physical property of strongly correlated electrons on a three-dimensional (3D) 3 x 3 x 3 cluster of the simple cubic lattice is here presented.In the work we developed the unit step Hamiltonian as a solution to the single band Hubbard Hamiltonian for the case of two electrons interaction in a finite three dimensional lattice. The approximation to the Hubbard Hamiltonian study is actually necessary because of the strong limitation and difficulty pose by the Hubbard Hamiltonian as we move away from finite - size lattices to larger N - dimensional lattices. Thus this work has provided a means of overcoming the finite - size lattice defects as we pass on to a higher dimension. We have shown in this study, that the repulsive Coulomb interaction which in part leads to the strong electronic correlations, would indicate that the two electron system prefer not to condense into s-wave superconducting singlet state (s = 0), at high positive values of the interaction strength. This study reveals that when the Coulomb interaction is zero, that is, for free electron system (non-interacting), thevariational parameters which describe the probability distribution of lattice electron system is the same. The spectra intensity for on-site electrons is zero for all values of the interaction strength
Pulmonary Thromboembolism - etilogy, types, medical- Surgical and nursing man...VarunMahajani
Disruption of blood supply to lung alveoli due to blockage of one or more pulmonary blood vessels is called as Pulmonary thromboembolism. In this presentation we will discuss its causes, types and its management in depth.
micro teaching on communication m.sc nursing.pdfAnurag Sharma
Microteaching is a unique model of practice teaching. It is a viable instrument for the. desired change in the teaching behavior or the behavior potential which, in specified types of real. classroom situations, tends to facilitate the achievement of specified types of objectives.
Title: Sense of Taste
Presenter: Dr. Faiza, Assistant Professor of Physiology
Qualifications:
MBBS (Best Graduate, AIMC Lahore)
FCPS Physiology
ICMT, CHPE, DHPE (STMU)
MPH (GC University, Faisalabad)
MBA (Virtual University of Pakistan)
Learning Objectives:
Describe the structure and function of taste buds.
Describe the relationship between the taste threshold and taste index of common substances.
Explain the chemical basis and signal transduction of taste perception for each type of primary taste sensation.
Recognize different abnormalities of taste perception and their causes.
Key Topics:
Significance of Taste Sensation:
Differentiation between pleasant and harmful food
Influence on behavior
Selection of food based on metabolic needs
Receptors of Taste:
Taste buds on the tongue
Influence of sense of smell, texture of food, and pain stimulation (e.g., by pepper)
Primary and Secondary Taste Sensations:
Primary taste sensations: Sweet, Sour, Salty, Bitter, Umami
Chemical basis and signal transduction mechanisms for each taste
Taste Threshold and Index:
Taste threshold values for Sweet (sucrose), Salty (NaCl), Sour (HCl), and Bitter (Quinine)
Taste index relationship: Inversely proportional to taste threshold
Taste Blindness:
Inability to taste certain substances, particularly thiourea compounds
Example: Phenylthiocarbamide
Structure and Function of Taste Buds:
Composition: Epithelial cells, Sustentacular/Supporting cells, Taste cells, Basal cells
Features: Taste pores, Taste hairs/microvilli, and Taste nerve fibers
Location of Taste Buds:
Found in papillae of the tongue (Fungiform, Circumvallate, Foliate)
Also present on the palate, tonsillar pillars, epiglottis, and proximal esophagus
Mechanism of Taste Stimulation:
Interaction of taste substances with receptors on microvilli
Signal transduction pathways for Umami, Sweet, Bitter, Sour, and Salty tastes
Taste Sensitivity and Adaptation:
Decrease in sensitivity with age
Rapid adaptation of taste sensation
Role of Saliva in Taste:
Dissolution of tastants to reach receptors
Washing away the stimulus
Taste Preferences and Aversions:
Mechanisms behind taste preference and aversion
Influence of receptors and neural pathways
Impact of Sensory Nerve Damage:
Degeneration of taste buds if the sensory nerve fiber is cut
Abnormalities of Taste Detection:
Conditions: Ageusia, Hypogeusia, Dysgeusia (parageusia)
Causes: Nerve damage, neurological disorders, infections, poor oral hygiene, adverse drug effects, deficiencies, aging, tobacco use, altered neurotransmitter levels
Neurotransmitters and Taste Threshold:
Effects of serotonin (5-HT) and norepinephrine (NE) on taste sensitivity
Supertasters:
25% of the population with heightened sensitivity to taste, especially bitterness
Increased number of fungiform papillae
Title: Sense of Smell
Presenter: Dr. Faiza, Assistant Professor of Physiology
Qualifications:
MBBS (Best Graduate, AIMC Lahore)
FCPS Physiology
ICMT, CHPE, DHPE (STMU)
MPH (GC University, Faisalabad)
MBA (Virtual University of Pakistan)
Learning Objectives:
Describe the primary categories of smells and the concept of odor blindness.
Explain the structure and location of the olfactory membrane and mucosa, including the types and roles of cells involved in olfaction.
Describe the pathway and mechanisms of olfactory signal transmission from the olfactory receptors to the brain.
Illustrate the biochemical cascade triggered by odorant binding to olfactory receptors, including the role of G-proteins and second messengers in generating an action potential.
Identify different types of olfactory disorders such as anosmia, hyposmia, hyperosmia, and dysosmia, including their potential causes.
Key Topics:
Olfactory Genes:
3% of the human genome accounts for olfactory genes.
400 genes for odorant receptors.
Olfactory Membrane:
Located in the superior part of the nasal cavity.
Medially: Folds downward along the superior septum.
Laterally: Folds over the superior turbinate and upper surface of the middle turbinate.
Total surface area: 5-10 square centimeters.
Olfactory Mucosa:
Olfactory Cells: Bipolar nerve cells derived from the CNS (100 million), with 4-25 olfactory cilia per cell.
Sustentacular Cells: Produce mucus and maintain ionic and molecular environment.
Basal Cells: Replace worn-out olfactory cells with an average lifespan of 1-2 months.
Bowman’s Gland: Secretes mucus.
Stimulation of Olfactory Cells:
Odorant dissolves in mucus and attaches to receptors on olfactory cilia.
Involves a cascade effect through G-proteins and second messengers, leading to depolarization and action potential generation in the olfactory nerve.
Quality of a Good Odorant:
Small (3-20 Carbon atoms), volatile, water-soluble, and lipid-soluble.
Facilitated by odorant-binding proteins in mucus.
Membrane Potential and Action Potential:
Resting membrane potential: -55mV.
Action potential frequency in the olfactory nerve increases with odorant strength.
Adaptation Towards the Sense of Smell:
Rapid adaptation within the first second, with further slow adaptation.
Psychological adaptation greater than receptor adaptation, involving feedback inhibition from the central nervous system.
Primary Sensations of Smell:
Camphoraceous, Musky, Floral, Pepperminty, Ethereal, Pungent, Putrid.
Odor Detection Threshold:
Examples: Hydrogen sulfide (0.0005 ppm), Methyl-mercaptan (0.002 ppm).
Some toxic substances are odorless at lethal concentrations.
Characteristics of Smell:
Odor blindness for single substances due to lack of appropriate receptor protein.
Behavioral and emotional influences of smell.
Transmission of Olfactory Signals:
From olfactory cells to glomeruli in the olfactory bulb, involving lateral inhibition.
Primitive, less old, and new olfactory systems with different path
- Video recording of this lecture in English language: https://youtu.be/lK81BzxMqdo
- Video recording of this lecture in Arabic language: https://youtu.be/Ve4P0COk9OI
- Link to download the book free: https://nephrotube.blogspot.com/p/nephrotube-nephrology-books.html
- Link to NephroTube website: www.NephroTube.com
- Link to NephroTube social media accounts: https://nephrotube.blogspot.com/p/join-nephrotube-on-social-media.html
Factory Supply Best Quality Pmk Oil CAS 28578–16–7 PMK Powder in Stockrebeccabio
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Hot Selling Organic intermediates
Prix Galien International 2024 Forum ProgramLevi Shapiro
June 20, 2024, Prix Galien International and Jerusalem Ethics Forum in ROME. Detailed agenda including panels:
- ADVANCES IN CARDIOLOGY: A NEW PARADIGM IS COMING
- WOMEN’S HEALTH: FERTILITY PRESERVATION
- WHAT’S NEW IN THE TREATMENT OF INFECTIOUS,
ONCOLOGICAL AND INFLAMMATORY SKIN DISEASES?
- ARTIFICIAL INTELLIGENCE AND ETHICS
- GENE THERAPY
- BEYOND BORDERS: GLOBAL INITIATIVES FOR DEMOCRATIZING LIFE SCIENCE TECHNOLOGIES AND PROMOTING ACCESS TO HEALTHCARE
- ETHICAL CHALLENGES IN LIFE SCIENCES
- Prix Galien International Awards Ceremony
Flu Vaccine Alert in Bangalore Karnatakaaddon Scans
As flu season approaches, health officials in Bangalore, Karnataka, are urging residents to get their flu vaccinations. The seasonal flu, while common, can lead to severe health complications, particularly for vulnerable populations such as young children, the elderly, and those with underlying health conditions.
Dr. Vidisha Kumari, a leading epidemiologist in Bangalore, emphasizes the importance of getting vaccinated. "The flu vaccine is our best defense against the influenza virus. It not only protects individuals but also helps prevent the spread of the virus in our communities," he says.
This year, the flu season is expected to coincide with a potential increase in other respiratory illnesses. The Karnataka Health Department has launched an awareness campaign highlighting the significance of flu vaccinations. They have set up multiple vaccination centers across Bangalore, making it convenient for residents to receive their shots.
To encourage widespread vaccination, the government is also collaborating with local schools, workplaces, and community centers to facilitate vaccination drives. Special attention is being given to ensuring that the vaccine is accessible to all, including marginalized communities who may have limited access to healthcare.
Residents are reminded that the flu vaccine is safe and effective. Common side effects are mild and may include soreness at the injection site, mild fever, or muscle aches. These side effects are generally short-lived and far less severe than the flu itself.
Healthcare providers are also stressing the importance of continuing COVID-19 precautions. Wearing masks, practicing good hand hygiene, and maintaining social distancing are still crucial, especially in crowded places.
Protect yourself and your loved ones by getting vaccinated. Together, we can help keep Bangalore healthy and safe this flu season. For more information on vaccination centers and schedules, residents can visit the Karnataka Health Department’s official website or follow their social media pages.
Stay informed, stay safe, and get your flu shot today!
Anti ulcer drugs and their Advance pharmacology ||
Anti-ulcer drugs are medications used to prevent and treat ulcers in the stomach and upper part of the small intestine (duodenal ulcers). These ulcers are often caused by an imbalance between stomach acid and the mucosal lining, which protects the stomach lining.
||Scope: Overview of various classes of anti-ulcer drugs, their mechanisms of action, indications, side effects, and clinical considerations.
These simplified slides by Dr. Sidra Arshad present an overview of the non-respiratory functions of the respiratory tract.
Learning objectives:
1. Enlist the non-respiratory functions of the respiratory tract
2. Briefly explain how these functions are carried out
3. Discuss the significance of dead space
4. Differentiate between minute ventilation and alveolar ventilation
5. Describe the cough and sneeze reflexes
Study Resources:
1. Chapter 39, Guyton and Hall Textbook of Medical Physiology, 14th edition
2. Chapter 34, Ganong’s Review of Medical Physiology, 26th edition
3. Chapter 17, Human Physiology by Lauralee Sherwood, 9th edition
4. Non-respiratory functions of the lungs https://academic.oup.com/bjaed/article/13/3/98/278874
ARTIFICIAL INTELLIGENCE IN HEALTHCARE.pdfAnujkumaranit
Artificial intelligence (AI) refers to the simulation of human intelligence processes by machines, especially computer systems. It encompasses tasks such as learning, reasoning, problem-solving, perception, and language understanding. AI technologies are revolutionizing various fields, from healthcare to finance, by enabling machines to perform tasks that typically require human intelligence.
TEST BANK for Operations Management, 14th Edition by William J. Stevenson, Ve...kevinkariuki227
TEST BANK for Operations Management, 14th Edition by William J. Stevenson, Verified Chapters 1 - 19, Complete Newest Version.pdf
TEST BANK for Operations Management, 14th Edition by William J. Stevenson, Verified Chapters 1 - 19, Complete Newest Version.pdf
Lung Cancer: Artificial Intelligence, Synergetics, Complex System Analysis, S...Oleg Kshivets
RESULTS: Overall life span (LS) was 2252.1±1742.5 days and cumulative 5-year survival (5YS) reached 73.2%, 10 years – 64.8%, 20 years – 42.5%. 513 LCP lived more than 5 years (LS=3124.6±1525.6 days), 148 LCP – more than 10 years (LS=5054.4±1504.1 days).199 LCP died because of LC (LS=562.7±374.5 days). 5YS of LCP after bi/lobectomies was significantly superior in comparison with LCP after pneumonectomies (78.1% vs.63.7%, P=0.00001 by log-rank test). AT significantly improved 5YS (66.3% vs. 34.8%) (P=0.00000 by log-rank test) only for LCP with N1-2. Cox modeling displayed that 5YS of LCP significantly depended on: phase transition (PT) early-invasive LC in terms of synergetics, PT N0—N12, cell ratio factors (ratio between cancer cells- CC and blood cells subpopulations), G1-3, histology, glucose, AT, blood cell circuit, prothrombin index, heparin tolerance, recalcification time (P=0.000-0.038). Neural networks, genetic algorithm selection and bootstrap simulation revealed relationships between 5YS and PT early-invasive LC (rank=1), PT N0—N12 (rank=2), thrombocytes/CC (3), erythrocytes/CC (4), eosinophils/CC (5), healthy cells/CC (6), lymphocytes/CC (7), segmented neutrophils/CC (8), stick neutrophils/CC (9), monocytes/CC (10); leucocytes/CC (11). Correct prediction of 5YS was 100% by neural networks computing (area under ROC curve=1.0; error=0.0).
CONCLUSIONS: 5YS of LCP after radical procedures significantly depended on: 1) PT early-invasive cancer; 2) PT N0--N12; 3) cell ratio factors; 4) blood cell circuit; 5) biochemical factors; 6) hemostasis system; 7) AT; 8) LC characteristics; 9) LC cell dynamics; 10) surgery type: lobectomy/pneumonectomy; 11) anthropometric data. Optimal diagnosis and treatment strategies for LC are: 1) screening and early detection of LC; 2) availability of experienced thoracic surgeons because of complexity of radical procedures; 3) aggressive en block surgery and adequate lymph node dissection for completeness; 4) precise prediction; 5) adjuvant chemoimmunoradiotherapy for LCP with unfavorable prognosis.
2. 30 2 Mechanical Stress Order of Grain Boundaries
Crystal I Crystal II
Z
SII
Y
X
SI
Fig. 2.1 Diagram explaining how lattices of crystals I and II are generated from a reference system
XYZ by the transformations SI and SII, respectively
For a 2D distribution, this approach is equivalent to the Kröner generalized 3D
approach in which the continuous distribution of dislocations is expressed by a dis-
location density tensor α related to the β distortion by [3, 4]:
curl β = α (2.2)
The expression (2.1) reformulated by Christian in matrix notation leads to the
equation known as the Frank-Bilby equation that gives the Burgers vector density B
necessary to realise the compatibility at the interface between the crystals I and II:
B = (S−1
I − S−1
II ) · X (2.3)
SI and SII are affine transformations that generate the lattices of crystals I and II
starting from a reference system (Fig.2.1), X is a vector of relatively large intensity
in the grain boundary plane and B is the Burgers vector content of all the dislocations
that are crossed by the vector X. If this equation is satisfied, the grain boundary is
free of long-range stresses.
Equation(2.3) is relevant for any interface but we focus here on grain boundaries.
For grain boundaries in cubic materials, a simple relation links the SI and SII
lattices to the rotation R between the crystals: R = SII.S−1
I . Equation(2.3) then
becomes equivalent to the 0-lattice equation in which I is the identity matrix:
B = (I − R−1
)X (2.4)
In Eqs.(2.3) and (2.4), B is not considered as the sum of discrete Burgers vectors
belonging to the crystal translation lattice or to the DSC lattice. Such a discretization
depends on the relaxation processes and is meaningless when the dislocation density
is very high (Fig.2.2).
B may be determined by using the Frank circuit drawn around the grain boundary,
equivalent to the Burgers circuit within the crystal (Fig.2.2). A vector X = XII,
traced in the grain boundary plane and indexed in lattice II, is obtained by applying
3. 2.1 Continuous Approach: The Frank-Bilby Equation 31
Fig. 2.2 Frank circuit: a
vector XI = XII is drawn in
lattice I, then surrounded by
a closed circuit starting and
finishing at the vector origin.
The same circuit is reproduced
around the grain boundary: it
starts at the extremity of XII
(vector in the grain boundary
plane obtained by the rotation
θ of the vector (XI) passes
through the origin O and
terminates at the extremity
of XI. This circuit displays
a closure failure B given by
Eq.(2.4) by substituting X for
XII
the rotation R to a vector XI in lattice I:
XI = R−1
XII.
In the cases of symmetrical grain boundaries, if a median lattice between the two
crystal lattices is considered as the reference system, then the I and II lattices are
deduced from this reference by equal and opposite rotations (± θ/2) around a ρ axis;
expression (2.4) takes the well-known form of the Frank formula:
B = 2 sin(θ/2)(X ∧ ρ) (2.5)
And provided that X is perpendicular to the rotation axis:
B = 2 sin(θ/2)|X| (2.6)
The absence of long-range stresses around the interface may be modelled by the
use of two continuous distributions of dislocations with Burgers vector densities of
opposite sign that cancel. One distribution may be regarded as an array of stress-
generator dislocations or coherency dislocations, the other as an array of stress-
annihilator dislocations or anti-coherency dislocations.For a detailed description of
this approach, the reader may refer to the book by Sutton and Balluffi [8] and to the
articles by Olson and Cohen [9] and Bonnet [10, 11].
4. 32 2 Mechanical Stress Order of Grain Boundaries
2.2 Discrete Approach: The Read and Shockley Model
Studies of edge dislocation walls in a crystal, later recognised as a low-angle sym-
metrical tilt grain boundaries, developed about 20 years before the explicit structural
model of these grain boundaries due to Read and Shockley [7]. This model links the
grain boundary misorientation angle θ to the spacing d between the edge disloca-
tions of Burgers vector b that form a periodic network in the grain boundary plane
(Fig.2.3):
d =
b
θ
(2.7)
Fig. 2.3 Scheme showing
a low angle symmetrical tilt
grain boundary according to
Read and Shockley. The tilt
axis is perpendicular to the
plane of the figure
The classical Read and Shockley formula gives the intergranular energy in func-
tion of the misorientation angle, in the low-angle limit:
γθ = γθ θ(A − lnθ) (2.8)
γ0 = μb/4π(1 − ν) and A = b/2πr0 with μ, the shear modulus, ν the Poisson
coefficient and r0 the dislocation core radius.
Equation(2.8) is easily obtained by summing the elastic energies of the N
edge dislocations (of unit length) contained in a unit area of the grain boundary
(N = 1/d). This calculation may be done provided the dislocations are sufficiently
far from each other (low θ angle) and the extension of the stress fields is equal to the
distance d. This equation, initially established for θ < 15◦, is fundamental. Indeed,
after having established their model, Read and Shockley predict its relevance for
high-angle grain boundaries formed by dislocations evenly spaced. In 1989, Wolf
proposed an empirical extrapolation of the Read and Shockley formula for high-angle
grain boundaries sharing the same mean plane, by substituting θ by sin θ [12].
In1947,LacombeandBeaujard[13]revealedtheexistenceofsub-grainboundaries
in aluminium single crystals owing to the preferential etching of the crystal surface
on the form of etch pit alignments. But no relationship was established between etch
pits and dislocations. In 1950, Read and Shockley suggested a correlation between
5. 2.2 Discrete Approach: The Read and Shockley Model 33
their model and the previous observations. It was only in 1953 that a proper verifica-
tion of the validity of the dislocation model for sub-grain boundaries was obtained
on a germanium bicrystal by Vogel et al. [14].
2.3 Bollmann’s Discrete Approach: Intrinsic Dislocations
The Bollmann approach generalizes the Read and Shockley model of intergranular
dislocations at any grain boundary. It rests on Eqs.(1.23) and (1.24) that define the
O-lattice and the 02-lattice. The term intrinsic, which comes from the Latin intrin-
secus, indicates that these dislocations are inherent in the grain boundary structure
that cannot exist in their absence; they actually are structural dislocations.
2.3.1 Primary Intrinsic Dislocations
In the 0-lattice equation (I − R−1) xO = Bm, the Bm vector may always be con-
sidered as the sum of Burgers vectors bm of crystal I (subscript m for matrix).
Equations(1.23) and (1.22) for non-cubic materials) may be seen as the quantified
form of the Frank-Bilby Eq.(2.4) for grain boundaries in which x0 is a O-lattice
element. The different O elements are surrounded by Wigner-Seitz cell walls. The
intersections of the grain boundary plane with these walls form the b-net; they define
the lines of the geometrically necessary dislocations called primary intrinsic dislo-
cations (IGBD) localized in the regions of worse fit between the two crystals. The
term primary indicates that these dislocations account for the deviation from the first
invariant of the bicystal, i.e. the single crystal (θ = 0). The term secondary will be
used to explain the deviation from the second invariant, the CSL lattice.
The primary intrinsic dislocations possess a Burgers vector bm of the perfect
crystal dislocations and form a network with the same periodicity than the 0-lattice
(Fig.2.4). Their spacing decreases when the misorientation angle increases according
to the formula derived from the Frank Eq.(2.3) where |X|= d:
d =
bm
2sin θ/2
(2.9)
For low θ values, Eq.(2.9) is identical to the Read and Shockley expression (2.7).
The uniformly spaced distribution of the primary dislocations mathematically
predicted by Eq.(2.9) is generally not respected. Indeed, a spacing between two
dislocations must be physically compatible with the lattice period. Only the average
spacing measured on an arrangement of several dislocations meet this formula.
Let us illustrate the application of Eq.(2.4) to the prediction of the primary dislo-
cation content in a tilt grain boundary, then in a twist grain boundary, both displaying
a small misorientation θ = ε.
6. 34 2 Mechanical Stress Order of Grain Boundaries
Fig. 2.4 Diagram showing the localization of the primary intrinsic dislocations in a low angle tilt
grain boundary. A, B and O are nodes of the O-lattice. PI and PII (QI and QII) are homologous nodes
assimilated to one node. D represents a region of bad fit where the primary dislocation is localized
• For a tilt grain boundary around [011] with θ = ε, cos ε # 1 and sin ε # ε. The
rotation matrix R between the crystals is:
R =
1 − ε ε
ε 1 0
− ε 0 1
(2.10)
I − R−1
=
0 − ε ε
ε 0 0
− ε 0 0
(2.11)
The indices of any vector in the (01−1) grain boundary plane are [xyy].
The product (I − R−1)X in Eq.(2.4) enables to define the three components of the
dislocation density vector B:
Bx = 0 By = εx Bz = − εx (2.12)
B is then perpendicular to the rotation axis and to the grain boundary plane (Fig.2.5)
Furthermore:
B = nbm and bm = a/2 [01 − 1]. (2.12bis)
The dislocations have an edge character.
There are exact solutions for relations (2.4) only for vectors X = x0 = [xyy]
such as:
x = ± bm/ ε x = ±2 bm/ ε x = ± n bm/ ε (2.13)
The X vectors cut the intergranular dislocations along [100], the dislocation lines
perpendicularto[100]inthegrainboundaryplaneareparallelto[011].Thelow-angle
tilt grain boundary is well described by edge dislocations parallel to the rotation axis
7. 2.3 Bollmann’s Discrete Approach: Intrinsic Dislocations 35
Fig. 2.5 Geometry of the
primary dislocation network
(b = a/2 [01−1]) in a low angle
[011] tilt grain boundary
and distant from d = b/θ, as predicted by Read and Shockley. In this description,
the distance between the dislocations is large as θ = ε is a very small angle.
We may also describe the low angle tilt boundary by using an equivalent rotation
between the crystals: θ = (ε +π), then cos (ε +π)# −1 and sin(ε +π) − ε. The
application of the previous procedure leads to predict the occurrence of dislocations
also parallel to the tilt axis but with a Burgers vector b = a [100]. Furthermore,
the distance between the dislocations is now very small ( θ = ε +π being large).
As an example, for a 2◦ misorientation in copper, the dislocations are distant from
about 7nm according to the first description; they may be observed by conventional
transmission electron microscopy. According to the second description, the distance
is about 0.12nm and the dislocations are no longer individually distinguishable; the
physical reality of this description in terms of dislocations is questionable. Rather in
this case to use a continuous distribution of the deformation. It seems reasonable to
consider that the dislocation structure predicted by the first description is likely to
be realised. This point will be discussed further.
Figure2.6 presents a primary dislocation network in a low-angle tilt grain bound-
ary (2◦ around 111) in a iron–molybdenum alloy with a b.c.c. structure [15].
The same approach for a low-angle twist grain boundary with a {001} plane results
in a structure composed of two screw dislocation networks perpendicular to each
other and with a Burgers vector b = a/2 110 ; the distances between the dislocations
Fig. 2.6 Electronic micro-
graph of a 2◦ 111 tilt grain
boundary in a Fe-1% Mo
showing a primary edge dislo-
cation network [15]
8. 36 2 Mechanical Stress Order of Grain Boundaries
Fig. 2.7 Electronic micrograph of a low-angle twist grain boundary θ# 4◦ around 001 , in gold
showing two primary screw dislocation networks aligned along the 110 directions (at 45◦ of g);
the spacing between the dislocations is d# 40nm [16]. Bubbles are artefacts (bad welded bicrystal
regions) resulting from the bicrystal manufacturing; disturbed parts of the network (as EF) indicate
the presence of extrinsic dislocations (see Sect.5.2)
agree with the experiments (Fig.2.7) [16]. As previously, other descriptions exist but
their physical meaning is questionable.
It must be stressed that, whatever the chosen description of a given grain boundary,
the dislocation content totally accounts for the rotation between the crystals: these
dislocations are geometrically necessary. But the adopted grain boundary structure
depends on the relaxations, and only the observation enables to specify what descrip-
tion is physically meaningful.
Besides, the distance between the dislocations decreases as the misorientation
increases and becomes so small that it is no longer compatible with the crystal lattice
periodicity; the dislocation stress fields then overlap. However, the O-lattice being a
continuous function of the misorientation, the primary dislocations must exist from
a strict geometrical point of view. This continuity is confirmed by the observation
of an electron (or X-ray synchrotron) diffraction pattern of a grain boundary plane
[17]. Indeed, the periodic distribution of deformations in the plane constitutes an
optic grid; by selecting a diffraction spot, an electron microscopy image of the grid
may be obtained. The distance between the rows of this rebuilt network well satisfies
Eq.(2.9).
Insummary, theequationgivingthetotal dislocationcontent B inagrainboundary
is mathematically valid whatever the misorientation θ; but, for high θ values, it covers
the physical concept of dislocation density. Another discrete description of the grain
boundary structure is then proposed which refers to the second invariant of the
bicrystal, i.e. the CSL lattice.
9. 2.3 Bollmann’s Discrete Approach: Intrinsic Dislocations 37
2.3.2 Secondary Intrinsic Dislocation
As postulated by Bollmann, any grain boundary adopts as much as possible the struc-
ture of the neighboured coincidence grain boundary, provided relaxations on the form
of secondary intrinsic dislocations. These dislocations are localized in the regions
of bad fit between the two DSC lattices of the coincidence grain boundary θc rotated
one with respect to other by an angle equal to the angular deviation between the real
and the coincidence grain boundaries. These lines correspond to the intersections of
the grain boundary plane with the walls of the O2-lattice predicted by Eq.(1.24).
The term secondary comes from the fact that these defects break the periodicity
of the primary dislocations. They form a sub-grain boundary in the grain boundary.
Then, coincidence grain boundary regions delimited by secondary dislocations con-
stitute the grain boundary (Fig.2.8).
Fig. 2.8 Schematic representation of the periodic arrangement of secondary dislocations (⊥) in
the primary network (⊥)
Conversely to the plurality of the primary dislocation structures for a given grain
boundary, only one secondary dislocation structure is predicted by Eq.(1.24). Indeed,
whatever the equivalent rotations chosen for the real and the coincidence grain
boundaries, the angular deviation matrix D is identical. However, different struc-
tures may exist for non-mathematical reasons, linked to the atomic relaxations in the
boundary.
The characteristics of the secondary dislocations depend on the tilt/twist character
of the deviation to the coincidence. In a tilt grain boundary, a pure tilt deviation may
exist either around the fundamental rotation axis or around another axis; otherwise
10. 38 2 Mechanical Stress Order of Grain Boundaries
the deviation may have a twist or mixed character. Depending on the case, the grain
boundary structure is constituted of only one secondary edge dislocation network
or of two screw dislocation networks or even several mixed dislocation networks.
Similarly, a twist grain boundary may present a twist, tilt or mixed angular deviation
to the coincidence. In any case, the dislocation lines correspond to the intersections
of the grain boundary plane with the O2-lattice walls.
The Burgers vectors of the secondary dislocations belong to the DSC lattice, but
are not necessarily the elemental vectors of this lattice. In order to determine the
Burgers vector of a secondary dislocation, a closed circuit drawn around the defect
in the grain boundary must be reproduced in the DSC lattice of the nearest coincident
grain boundary. A construction very useful for analysing the high-resolution trans-
mission electron microscopy images enables to determine the dislocation Burgers
vector and its associated step (Fig.2.9) [18]. A secondary dislocation provokes a
translation of the CSL lattice. A step vector s may be defined, that relates an initial
coincidence node to the coincidence node obtained by translation. If the crystal I is
fixed and the crystal II translated, then the step sIis defined in the coordinate system
of crystal I; and reciprocally for sIIif the crystal II is fixed. There are several choices
for the step vectors, related to each other by a DSC vector; the shortest one is usually
chosen. The height of the step associated to the dislocation is hI = sI n in crystal I
and hII = sII n in crystal II. It follows that:
b = sI − sII
h = n(sI + sII)/2 (2.14)
where n is the normal to the grain boundary plane (pointed from crystal I to crystal II).
The Burgers vectors of the secondary dislocations must be necessarily indexed in
the reference system of the bicrystal translation lattice (the DSC lattice). Otherwise,
if each vector si is indexed in the reference system of crystal i, we obtain the primary
Burgers vector of the defect.
Fig. 2.9 Construction in the DSC lattice of a = 5 grain boundary of three possible steps s, s and
s associated to a dislocation with vector ± b1. The empty and full symbols correspond to crystal
II and crystal I, respectively; the triangular symbols represent sites located at ± a/2 [100] along the
projection axis; a CSL in its original position; b Displacement of crystal II relative to crystal I by
a vector + b1; c displacement of crystal I relative to crystal II by a vector − b1 [18]
11. 2.3 Bollmann’s Discrete Approach: Intrinsic Dislocations 39
The average distance between the secondary dislocations is given by a formula
similar to (2.9)
d =
|bDSC|
2sin θ/2
(2.15)
This distance decreases when the coincidence index increases (|bDSC| decreases)
and/or when the angular deviation θ to the coincidence increases. As a result, the
dislocations cannot be experimentally distinguished beyond a certain value or
beyond a certain θ deviation from the neighbouring coincidence relationship. Fur-
thermore, with the diminution of |bDSC|, the contrasts associated to dislocations in
transmission electron microscopy decreases. Thus, while preserving an almost con-
stant spacing, the dislocations are less and less visible when increases. Figure2.10
illustrates this effect for a series of 001 twist grain boundaries in gold [19].
Fig. 2.10 Intrinsic secondary dislocation networks in 001 twist grain boundaries in gold showing
the diminution of their contrast when increases: a = 1, b = 0,289nm; b = 5, b = 0,129nm;
c = 13, b = 0,080nm; d = 17, b = 0,070nm [19]
12. 40 2 Mechanical Stress Order of Grain Boundaries
Figure2.11 presents a network of secondary dislocations in a tilt grain boundary
in nickel that totally account for the deviation θ = 0, 09◦ 112 of this boundary
with respect to the = 3 {111} coincidence grain boundary [20]. Dislocations A
are parallel to the 112 direction, their Burgers vector is the DSC vector of = 3
normal to the grain boundary plane, b = a/3 111 , and their distance (≈ 130nm) is
in good agreement with the calculated distance.
Fig. 2.11 Electronic micrograph in dark field (crystal C2 is in extinction condition) showing a
network of intrinsic secondary dislocations (as A) in a near = 3 {111} tilt grain boundary in
nickel (B is an extrinsic dislocation that will be analysed in part II, Chap.5) [20]
The near one-dimensional coincidence grain boundaries displays a network of
secondary dislocations, the lines of which being parallel to the intersections of the
crystal planes in good fit with the grain boundary plane.
In an approximate coincidence grain boundary, the structure in terms of dis-
location must account for the rotation and the deformation necessary to bring in
coincidence two multiple cells of the crystals. According to the choice of the rational
value of the (c/a)2 ratio that approximates the real value, these two components
(rotation and deformation) differ. Only the observed intergranular structure enables
to specify the relaxation mode chosen by the grain boundary. Let us consider a grain
boundary near the rhombohedral twin, often observed in alumina or hematite poly-
crystals. When the grain boundary plane is strictly parallel to the {01-12} twin plane,
the grain boundary is described by an exact bi-dimensional coincidence. But, most
often, the observed grain boundary deviates from {01-12}. Then, the grain boundary
is described in terms of tri-dimensional approximate coincidence, by reference to
specific rotations θ around (02-21) associated to rational values of (c/a)2 as close
as possible of the real value, (2.730)2 for alumina. The specific rotations with their
corresponding deformations ε for alumina are (Table2.1) [21]:
Table 2.1 Specific rotations
and associated deformations
giving rise to rational
coincidences for rational
values of the c/a ratio [21]
c/a θ◦ ε
7 2.739 85.90 0.003
29 2.728 86.05 0.001
36 2.730 86.02 0
13. 2.3 Bollmann’s Discrete Approach: Intrinsic Dislocations 41
The deformations decrease when increases, this is also true for hematite
(c/a = 2,733). Among the three previous descriptions, which is the one adopted by
the grain boundary: high coincidence index ( = 36) and zero deformation or small
coincidence index ( = 7) at the expense of a deformation? Only the experiment
enables to answer the question.
The DSC vectors of the different descriptions are such that the intensities of two
of them, b2 and b3, are almost identical whereas the third vector b1 decreases as
increases. A secondary dislocation network may be observed in a grain boundary,
the misorientation of which being close to that of the rhombohedral twin in alumina
(θ = 85◦5); the grain boundary plane is deviated by 8◦ from the rhombohedral plane
(Fig.2.12). The Burgers vector of the intrinsic dislocations have been identified as
a linear combination of the two almost identical vectors (b2 and b3), whatever the
grain boundary description. In that case, no approximate coincidence description
seems to be preferred. However, in a grain boundary with a plane very close {01-12}
in hematite, the secondary dislocations, observed by high-resolution transmission
electron microscopy, possess a Burgers vector that is the smallest DSC vector cor-
responding to a = 7 coincidence relationship [22].
Fig. 2.12 Network of intrin-
sic secondary dislocations C
in a grain boundary near the
rhombohedral twin in alumina
[21] (Extrinsic dislocations B
will be described in part II)
In the θ = 90◦ (10-10) grain boundaries that are numerous in tungsten carbide of
hexagonal symmetry, the secondary dislocations have Burgers vectors of the = 2
coincidence relationship with a deformation equal to 0.024 whereas another neigh-
boured description with a = 39 coincidence index only requires a 0.0016 defor-
mation [23]. Similar conclusions have been obtained for the (01-12) twin in zinc of
hexagonal structure. To decide what approximate coincidence is better adapted to
the grain boundary structure, the dislocation lines are determined; they correspond
to the intersections of the grain boundary plane with the O2-lattice associated to
the smallest coincidence index, = 13 in this case. But the description requires a
non-negligible deformation, about 0.5% [24].
Thus, in non-cubic system, the structure of a grain boundary close to different
approximate coincidences seems to adopt that of a grain boundary with a specific
rotation corresponding to the rational coincidence with the smallest index at the
expense of a larger deformation of the coincidence cell.
14. 42 2 Mechanical Stress Order of Grain Boundaries
2.3.3 Interest and Limit of the Intrinsic Dislocation Model
To sum up, intrinsic primary dislocations with Burgers vectors of the crystals describe
the low-angle grain boundary structures. Similarly, a high angle coincidence grain
boundary only contains primary dislocations, but the close spacing of these disloca-
tions does not give them any physical reality. Finally, a high-angle non-coincident
grain boundary contains primary dislocations accounting for the nearest coincidence
structureandsecondarydislocationsthataccommodatethedeviationbetweenthereal
and the coincidence grain boundaries. Most often, only the secondary dislocations
are visualized as the distances between the primary dislocations are too small.
Figure2.13 shows the evolutions of the expected dislocation spacing in function
of the misorientation angle θ and in function of the deviation θ from different coin-
cidences for 001 symmetrical tilt grain boundaries in gold [25]. The points reported
on the curves indicate the spacing experimentally determined on the conventional
transmission electron microscopy images. The smaller the coincidence index, the
larger the deviation θ authorized to visualize secondary dislocations.
Fig. 2.13 Evolution of the spacing d (in nm) in function of the misorientation θ of the dislocations in
001 symmetrical tilt grain boundaries in gold. The full line and the dotted lines give the spacing of
the primary and secondary dislocations, respectively. The points indicate in which grain boundaries
periodic dislocations have been observed [25]
The possibilities to simultaneously observe primary and secondary dislocations
are limited to few experiments; it was the case in gold for a [001] tilt grain boundary
deviated from about 1◦ from the nearest coincidence = 13 {320} (Fig.2.14).
The primary dislocations with Burgers vector a/2 [110] are distant from 0.77nm
(calculated distance 0.75nm) and the secondary edge dislocations with DSC Burgers
vector a/13 [320] are distant from 7nm, in agreement with the theoretical distance
of 7.1nm [25].
The Bollmann formalism does not claim to describe the physical reality of a
grain boundary, but constitutes a possible approach to knowledge. Bollmann himself
insists that the use of the 0-lattices only permits a discrete selection of the possible
15. 2.3 Bollmann’s Discrete Approach: Intrinsic Dislocations 43
Fig. 2.14 Electronic micrograph of a symmetrical [001] tilt grain boundary near = 13 {320} in
gold (g = 200). The fine periodic contrasts are due to primary dislocations with Burgers vector a/2
[110], the large contrasts (arrows) are attributed to secondary dislocations with Burgers vector a/13
[320] [25]
displacement vectors B, characteristic of the continuous displacement field of the
transformation A. It is a first approximate model of linear relaxation in the grain
boundary plane where the fact that it is linear is pure convenience.
Moreover, whatever the expression used to describe the grain boundary structure
in terms of dislocations, Burgers vector density according to Frank-Bilby or discrete
dislocations according to Bollmann, there are a large number of transformations S
(or R) equivalent because of the symmetry operations of the crystals, i.e. depending
on the choice of the unit cell in the reference coordinate lattice. An example has been
given for the low-angle [011] tilt grain boundary (see Sect.2.3.1). The equivalent
matrices are obtained by multiplying SI, SII or R by a uni-modular matrix U corre-
sponding to an operation of the point group symmetry of lattices I or II. Thus, there
are a very large number of possible descriptions of the structure of a given grain
boundary. Bollmann favours a particular description called NNR for Nearest Neigh-
bour Relationship that corresponds to the minimal dislocation density by arguing
that it must be associated to an energy minimum (E proportional to b2) [26]. This
argument is based on the consideration of the long-range elastic strain fields, but
the local distortions due to intrinsic dislocations cancel at large distance. A grain
boundary at equilibrium only generates short-range strains; the strain distribution
depends on the intergranular relaxation modes and cannot be predicted by a geo-
metrical model. The Frank-Bilby and Bollmann equations are disconnected from the
grain boundary energy. Finally, a grain boundary is a unique physical object with a
precise structure and a precise energy. Among all the possible structures, none can be
selected a priori. This question will be discussed more thoroughly in the description
of grain boundaries constrained at triple junctions (see Sect.10.2.2) where it takes a
particular importance.
Despite its geometrical nature, the Bollmann model has been successfully applied
to identify intrinsic dislocations in numerous grain boundaries in different types of
materials. It is especially remarkable that periodic dislocations have been observed,
by high-resolution transmission electron microscopy, in symmetrical tilt grain
boundaries with high coincidence indices, as = 337 in silicon; in that case,
16. 44 2 Mechanical Stress Order of Grain Boundaries
the Burgers vectors of the intrinsic secondary dislocations do not correspond to the
elemental DSC vectors [27].
A dislocation network always reveals the existence of a preferred ordered state of
the grain boundary, whatever its description. The reason why a preferred state exists
is a subject of questioning in solid-state physics.
However, the high-coincidence index grain boundaries displaying periodic dislo-
cations are not true general grain boundaries; they generally display misorientations
around axes with simple indices and possess symmetrical grain boundary planes.
What happens for grain boundaries with random misorientations and asymmetrical
high index planes? These grain boundaries do not present dislocation networks at
the scale of the conventional electron microscopy and are not observable by high-
resolution electron microscopy. Do they obey a discrete or continuous description? A
partial answer to this question has been advanced when we deal with the grain bound-
ary structure at high temperature (See Sect.4.1), then with relaxation phenomena of
non-equilibrated grain boundaries (See Chap.9).
In summary, the dislocations predicted by Eqs.(1.23) and (1.24) are geometrically
necessary, but the actual dislocation content depends on the local intergranular stress
relaxations. The structure is governed by the energy minimization and any forecast
based on geometry alone cannot reflect reality. It is physics that determines the
real mode of relaxation and a physical description requires an atomistic approach
(Chap.3).
2.4 Partial Intergranular Dislocations
The grain boundaries may also contain partial dislocations, generally associated to
rigid body translations τ, thus not predicted by the geometrical formalism. Although
non-periodic, these dislocations belong to the grain boundary structure [28]. Their
Burgers vectors are determined by using the circuit mapping method proposed by
Pond [29]. This circuit is analogous to a Frank circuit but is drawn in the dichromatic
complex; it takes into account all the symmetry operations of the space group and
not only the translations. Partial dislocations may also result from the existence of
grain boundary structure variants. If regions of different structures along the grain
boundary are related by a symmetry operation, they are energetically degenerate.
Partial dislocations separating such regions have been observed in germanium [30]
and in a copper–silicon alloy [31]. Partial dislocations are often located at the junction
between two boundary facets, but may be also present in a straight part of the grain
boundary. They may result of the decomposition of the DSC intrinsic dislocations, as
observed in a = 3 {211} grain boundary where a perfect dislocation with Burgers
vector a/3 111 dissociates into two products with Burgers vectors a/9 111 and
2a/9 111 [28].
17. 2.5 Stress Fields Associated to Intrinsic Dislocations 45
2.5 Stress Fields Associated to Intrinsic Dislocations
The stress field of a low-angle grain boundary has been calculated by Hirth and
Lothe, using isotropic linear elasticity and by adding the stress fields of the periodic
dislocations [32]. The dislocations are mixed; three components of their Burgers
vector (bx by bz) and thus three periodic networks are considered (Fig.2.15).
Fig. 2.15 The three periodic networks of dislocations that are considered to calculate the elastic
stress fields of a low angle grain boundary. The Burgers vector of the mixed dislocations is decom-
posed in two edge components along Ox (a), Oy (b) and one screw component along Oy (c) [32]
For a symmetrical tilt grain boundary constituted of egde dislocations with
Burgers vector bx, parallel to the z axis and distant from D along Oy (Fig.2.15a),
the stress components (in regions such as x D/2π) exponentially decrease with
the distance x from the grain boundary ; on the contrary the stress field of an isolated
dislocation is inversely proportional to the distance from its line. The expressions of
the shear component σxy for the two types of defects may be compared:
• for a dislocation
σxy =
μb
2π(1 − ν)
x(x2 − y2)
(x2 + y2)
(2.16)
• for a tilt grain boundary
σxy =
μb
(1 − ν)
2πx
D
exp
−2πx
D
cos
2πy
D
(2.17)
where, μ is the shear modulus and ν the Poisson coefficient.
The expression of σxx for the grain boundary is similar to that of σxy with
sin2πy/D instead of cos2πy/D. The value of x = 2D (y = 0), the shear compo-
nent of σxy due to the periodic arrangement, is about 1% of its value for an isolated
dislocation. In the grain boundary plane (x = 0) and for distances y < D/2π of a given
18. 46 2 Mechanical Stress Order of Grain Boundaries
dislocation in the network, more than 90% of the grain boundary stress σxxis due to
the dislocation at the origin. Midway between the two dislocations, the compression
stress of one dislocation is annihilated by the tension stress of the other. Then, there
is no long-range stress field induced by a symmetrical tilt grain boundary. If now
we consider the stress field due to a network of edge dislocations with a Burgers
vector by (Fig.2.15b), the component σyy does not cancel at long distance, in contra-
diction with the equilibrium condition given by Eqs.(2.3) or (2.4). The equilibrium of
the grain boundary imposes that there is no total Burgers vector parallel to the grain
boundary plane associated to edge dislocations. For the last component of the Burgers
vector bz (Fig.2.15c), the dislocations have a twist character. For a uni-dimensional
arrangement of dislocations parallel to Oz, the shear component σyz exponentially
tends to zero with the distance to the grain boundary, but the stress component σxz
preserves non-negligible values at large distance. A simple arrangement of screw
dislocations is then incompatible with the equilibrium condition (2.4). Equilibrium
in a pure twist grain boundary requires the formation of two screw dislocation net-
works perpendicular to each other that result in a rotation around the normal x; the
stress fields of these networks annihilate.
More generally, three dislocation networks with non-coplanar Burgers vectors
account for the misorientation of a random grain boundary. For a grain boundary
that satisfies the Frank-Bilby formula, the elastic stress field analysis shows that the
combination of the three stress fields yields their annihilation, even if one of the
network generates long-range stresses. In the calculation, the elastic constants of the
crystals are used, whereas the elastic constants of the grain boundary may differ from
those of the crystals as found by simulation [33]. Although the elastic constant tensor
of a boundary is not known, these constants may strongly vary in the grain boundary
plane [34].
Another model for calculating the stress fields is based on the grain boundary
description in terms of disclinations [35]. On the contrary to the dislocation model,
it takes into account the disclination core energy and may apply to high-angle grain
boundaries (see Sect.3.5).
More complex situations than those previously described by Hirth and Lothe
have been addressed, in particular by considering no strictly periodic networks and
by studying the evolution of their stress fields during return to periodicity. These
approaches will be detailed in part II, Chap.9 when dealing with the return to equi-
librium of a non-equilibrated grain boundary.
To conclude, the knowledge of the elastic contribution is insufficient to account
for the intergranular energy, we must know the contribution of the grain boundary
core. This constitutes the subject of the next session focussing on the grain boundary
atomic order.
19. References 47
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