physics of disordered materials - uppsala university€¦ · types of disordered solid materials...
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Physics of disorderedmaterials
Gunnar A. NiklassonSolid State Physics
Department of Engineering SciencesUppsala University
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Crystalline materials
• Introductory solid statephysics – physics of crystalline materials.
• Example: Si crystal used in semiconductor technology
• Effects of disorder importantfor many applications.
• Doping, nanoparticles, thinfilms
• Fundamental research
Source: Wikimedia, M. Lincetto
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Crystalline vs. amorphous solids
www.ces-world.com
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Course plan• Familiarity with the basic
description of disorderedstructures and associatedphysical concepts
• Giving neededbackground for followingresearch literature.
• Follows the layout of an introductory course basedon for example the book: C. Kittel: Introduction to Solid State physics
Outline• Structural models• Structural
characterization• Atomic vibrations
(phonons)• Electronic structure• Electrical properties• Optical properties• Magnetic properties
(P. Nordblad)
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Resources• Lecture notes• Reference books in the Solid State Physics
library, house 4, level 3.• Examination: Home assignments, 4 sets.• Contact: Gunnar Niklasson
Room 4412, phone: 3101. [email protected]
• Homepage: http://www.teknik.uu.se/ftf/education/Disordered_materials/Phys_disord_mtrl.html
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Weak disorder• No disorder: Chain,
ordered surface, crystal
• Weak disorder: Surface steps, defects, vacancies, dislocations, grain boundaries
• Perturbations of the perfect order
Schematic surface
Source: www1.columbia.edu
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Strong disorder• Homogeneous materials:
Local density similar to the average density.Ex: Glass
• Inhomogeneous materials: Large fluctuations of the local density, densitydepends on length scale, pores of different sizeEx: Aggregates, porousmaterials
Ni particles
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Types of disorder
• Topological (structural)No translationalsymmetry.
• Orientational (magneticspins, dipoles)
• Substitutional (chemical, compositional)
• Vibrational (atoms vibrate around theirequilibrium positions even at T=0.
Source: Elliott PoAM
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Inhomogeneous disorder• Structure repeats itself over
a range of length scales.• Lower cutoff: Atom,
molecule or particle size• Upper cutoff: Correlation
length, size of structure• Can often be described by
fractal geometry(B. Mandelbrot, 1975)
• Ex: Metallic particleaggregate, rough surface(von Koch curve)
Source: Wikipedia
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Ordering rule
• A property, p1, of one object in a set is related to pi of object ”i” by an ordering rule.
• Perfect order: pi of all objects can be derivedfrom knowing one of them, by applying the ordering rule.
• Crystalline order: Translational symmetry is the ordering rule and position is the property.
• Not only used for structural order but also order referred to different physical properties.
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Order parameter• Atoms A and B placed on
the sites of a crystallattice with concentrationscA and cB=1-cA.
• Crystal: All A-atoms are in A-sites (r=1) and all B-atoms in B-sites.
• Complete disorder: Atoms placed randomlyon the sites. Fraction cAof A-atoms in A-sites(r=cA).
• Order parameter
A
A
ccr
−−
=1
ξ
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Entropy and disorder
• Vacancies in a crystalMany arrangementspossible: W increases
• Configurational entropyΔS>0
• Mixing of two types of atoms or particles at constant volumeW increases
• Entropy of mixing >0
S = kB ln W
W – number of microscopic states of the system compatible with the macrostate.
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Types of disordered solid materials
• Amorphous solidsMany thin filmsa-semiconductorsa-metals
• Glasses (supercooledliquids)Vitreous silicachalcogenide glasses
• CompositesRandom alloysMetal-insulator mixturesParticle dispersionsPorous materials
• Often exhibit fractalstructure at intermediatelength scales
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Glass transition
• Rapid cooling of a liquid: Supercooled liquid• Not enough time to go into the lower energy
crystalline state at the melting temperature Tm
• As the temperature decreases below Tm the liquid becomes more viscous
• Forms a glass (a quenched amorphousstructure) in a region around a glass transitiontemperature Tg
• Tg depends on cooling rate and thermal history
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Glass transition temperature• Gradual change in
volume at Tg• Steep change in some
thermodynamicproperties (heat capacity, thermalexpansion, isothermalcompressibility)
• From liquid-like to solid-like values
• Determined by thermalmethods (Ex: DSC)
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Viscosity
• A solid has viscosity η~1013.6 Pa s• Ratio of shear stress to velocity gradient in a fluid
(”resistance to flow”)• η is proportional to the structural relaxation time, τ• Limit above is arbitrary corresponding to a relaxation
time of one day• Structural relaxation becomes very slow below Tg• Most simple liquids: 10-3 – 1 Pa s• Glass transition corresponds to 109 – 1012 Pa s,
depending on definition.
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Viscosity• Strong glass-formers:
Often covalent bonds, ex. SiO2. Arrhenius law:
• Fragile glass formers: Often ionic and organicmaterials. Vogel-Tammann-Fulcher law:
)/exp(0 TAηη =
))/(exp( 00 TTB −=ηη
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Glass transition• Tg for some materials:
SiO2 1453 KB2O3 530 KSe 313 KPMMA 378 KNylon-6 323 KPolyethylene 253 KPoly(ethyleneoxide) 218 KPoly (propyleneoxide) 211 K
• Physics still not known in detail
• Many theoretical ideas• Thermodynamics, free
volume theory, mode coupling theory…
• Connection to different relaxations, lowfrequency vibrations
• Subject stimulates boththeoretical thinking and novel experimental methods
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Origin of disorder
• Physical constraints:• Atoms and particles
may not penetrateeach other
• Chemical bonds• Interactions between
atoms and molecules• Attractive forces
between particles; v.d. Waals, dipolar…
Randomness Physical Laws
Disorder
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Example: DLA• Aggregation of particles
that move by a randomwalk
• Which physical law gives rise to aggregation?
• Particles stick to the growing aggregate due to short-range attractiveforces
• Such structures commonin physics and nature