Solidification

Driving force main equations

supercooling

∆𝐺 =  ∆𝐻 − 𝑇∆𝑆 = 𝐿 ∗ ∆𝑇/𝑇! metals: ∆𝑆 = 𝑅 inorganics: ∆𝑆 = 3𝑅

Homogenuos Nucleation in pure metals

- gain of energy: crystallization - need of energy: surface creation

∆𝐺! = −43𝜋𝑟!∆𝐺! + 4𝜋𝑟!𝛾!"

- volume difference liquid - solid: 2-4% - In order for nucleation to occur, average nucleus size must be larger than critical radius

Heterogenuos Nucleation in pure metals

Principle

Consequence for supercooling What about superheating?

Less curved surface needed, nucleation on already existing droplet Without heterogenuos nucleation: supercooling of over 100°C required. With heterogenuos supercooling: only couple of degrees. Superheating does not exist! Newly melted liquid does wet surface always, which means that ∆𝐺 = negative.

Solidification in pure metals

positive gradient = liquid warmer than solid negative gradient = liquid colder than solid

∆𝐺 ∝ 𝑈𝑚𝑤𝑎𝑛𝑑𝑙𝑢𝑛𝑔𝑠𝑑𝑟𝑢𝑐𝑘  𝑝   ∝ !!

Positive gradient: · ∆𝑆 < 2R: facetted growth (regular metals) · ∆𝑆 > 2.5R: non-facetted growth

(inorganics and semi metals

negative gradient: · dendritic growth for all pure metals as

outgrowth reaches into supercooled region = growth.

Solidification in alloys

Equilibrium Freezing

Freezing as we expect from phase diagram perfect mixing within solid and liquid, ie everywhere in the solid resp. everywhere in the liquid we get the same concentration Non equilibrium freezing

Scheil solidification general assumptions

Assumption: · uniform liquid composition · flat solid-liquid interface · No diffusion in the solid (fast cooling) · local equilibrium at interface Freezing: solid deprived from solute, accumulates in the melt. composition of solid = non-equilibrium eutectic can occur even if not expected!

𝑘 =𝑋!𝑋!

Scheil equations

𝑋! = 𝑋! ∗ 𝑓!

!!!!

𝑋! = 𝑘! ∗ 𝑋! ∗ (1 − 𝑓!)  !!!!

f = volume fraction solified only holds if densities of liquid and solid are similar and k is about constant.

Diffusional mixing of the liquid

Assumptions Consequence

Laminar flow, ie no mixing less mixing in the melt, no convective transport in the boundary layer, only diffusion SOLUTE BUILT UP local equilibrium at interface: 𝑘! =

!!"!!"

Cl,b: first increasing until diffusion from interface = solute rejected from solid = initial transistent.

Non equilibrium freezing, diffusional mixing:

what does … mean:

- k=1 - ke=k0

- k0<ke<1

- k=1: No or just little mixing of liquid

- ke=k0: No built up of solute

- k0<ke<1: Intermediate of the two

above

How to adjust mixing for purpose:

Maximum purification Rod of constant composition

Maximum purification: high mixing and low growth rate in order to get little built up. Constant composition: fast growth and little mixing in order to get large build up ( get over initial transistent)

Constitutional Supercooling

In pure metals: no change in solute concentration and hence a const. freezing temperature. In alloys with build up: freezing temperature changes according to solute concentration. If conc. decays faster and hence freezing temp. increases faster as the real temperature, even in a positive gradient! We get supercooling even in pos. gradients.

Constitutional supercooling

gradient and consequences

positive gradient: if conc. decay is fast enough, we get constitutional supercooling and hence dendritic growth If conc. decays slowly: flat growth negative gradient: dendritic growth, as for pure metals.

Hot tearing susceptibility

origin Prevention

eutectic alloys

Redistribution of alloy occurs at solidification front. Increased concentration in front of beta = higher supercooling = faster growth hinderence: creation of new surface area. optimum thickness = 2* minimal thickness

𝑣!"#  ∝  𝐷(∆𝑇)!

𝛾???

Influence of alloying elements on eutectic temperature

increasing: Cr, Si, W, Mo, Ti decreasing: Mn, Ni increased temperature = increased undercooling

Difference Recovery vs. Recrystallization

mechanism to store energy

increasing amount of stored energy

Recovery: creation of subgrains = Network of dislocations = Kleinwinkelkorngrenzen Recrystallization: direct creation of new, strainfree grains after cold work Mechanism to store energy: elastic strain, lattice defects (plastic deformation) Increasing amount of stored energy: higher purity, deformation, lower temp, grain size

Temperature of recovery mechanisms

generally, recovery starts at 0.3 Tm: low:

1. Migration of point defects to sinks 2. combination of point defects Intermediate: 1. Rearrangement of dislocation within tangles 2. Annihilation of dislocation, subgrain growth

high 1. dislocation climb, subgrain coalescence 2. Polygonization

static vs dynamic recovery

static recovery: recovery reliant on the heating of the existing, deformed microstructure dynamic recovery: recovery during deformation at elevated temperature

Mechanisms of recovery

1. Dislocation climb = movement of dislocations out of glide planes by interaction with vacancies.

2. Subgrain formation and growth = condensation of dislocations

3. Subgrain coalescence = disappearance of subgrain boundary during recovery = vanish of mismatch, small angle grain boundary becomes large angle

4. Polynoization = alignement of excess of edge dislocations created during strong deformation = large subgrains

factors affecting recovery

increasing recovery due to: higher temperature (dislocations rearrange faster) higher stacking fault energy (less disassociation of dislocations = more climb) less second phase particles (pin dislocations) less solute atoms (retardation due to solute drag)

Recrystallization

Definition places of nucleation

= nucleation and growth of strain free grains in certain areas of the material = strain free grains grow and consume deformed or recovered microstructure == primary/ discontinuous recrystallization: Nucleation and Growth Places: grain boundary, second phase particles, deformation bands

Driving forces in recrystallization latent deformation energy

grain boundary energy surface energy

1. latent deformation energy: Increase in dislocation density = increase in internal energy 𝑝 = 𝐺𝑏!∆𝜌   ≈ 100  !"

!! 2. Grain boundary energy:

each grain boundary carries a certain

amount of energy. 𝑝 = !!!!  (!"#$%.)

 ≈ 10   !"!!

3. Surface energy: exposure of certain crystallographic planes may increase internal energy

𝑝 = !∆!!!  (!"#$  !!!"#$%&&)

 ≈ 1   !"!!

Restricting forces in recrystallization Pressure on small grains second phase particles

surface grooves

1. Pressure on small grains: laplace pressure = reduction of high

curvature 𝑝 = !!!!  (!"#$%  !"#.)

≈ 10 − 100!"!!

2. second phase particles: particles at gb, takes away some of gb.

𝑝 = !  !! ∗!(!"#$%&'.!"#$%&'())  !!

≈ 10 − 100 !"!!

3. surface grooves: due to thermal etching, grooves

grow.            𝑝 = !!!

! !"#!!"#$%&  !!≈ 1 − 10 !"

!!

Balancing of forces Final grain size vs. second phase particle

Final grain size vs. sample size Final grain size vs. Critical deformation

Final grain size vs. second phase particles

grain growth stops at 𝐷 = !  (!"#$%!!"#$%&!!.)!  (!"#$%&'(')&*#)

= Zener Limit Final grain size vs. sample size grain growth stops at: 𝐷 = !!!

!!  𝑊   𝑑𝑟𝑎ℎ𝑡𝑑𝑖𝑐𝑘𝑒

Final grain size vs. critical deformation growing pressure larger than laplace pressure

𝐺𝑏!𝜌 >2𝛾!𝑅

Kinetics of primary recrystallization

Normal grain growth:

𝑣 =𝑑𝑅𝑑𝑡

=𝐷𝛼𝑅𝑇

𝑝                  𝑅 ∝ 𝑡 As growth sqrt(t) -> no single crystal possible Rate of Recrystallization Described by Johnson Mehl Avrami

𝑓 = 1 − exp −𝑘𝑡! , 𝑛 = 4,3

Secondary Recrystallization Description of Phenomena

Consequences (Microstructure) Requirements for it to happen

We always get slight differences in concentrations within the structure. At elevated temperature, second phase particles at grain boundaries (which inhibit grain growth by zener limit) may already dissolve Consequence: huge grains surrounded by small ones. Requirements: high degree of deformation, high annealing temperature

How is ETH Logo created?

1. stark verformtes Blech -> Rekrystallisation -> feines Korn

2. Lokale deformation = local erhöhte Versetzungsdichte = örtliche Rekristallisation zu grösseren Körnern

3. ätzen für sichtbar machen

Factors, which affect the recrystallization rate

temperature: thermal activation required heating rate: fast heating = less recovery = more recrystallization strain: more strain = more deformation = less recrystallization strain rate: higher strain rate= more rex initial grain size: small = more rex = small final grain (gb = nucleation site) second phase particles: large=promote rex (rex sites); small: retard rex (pinning effect) solute atoms: more solute= more pinning texture: heterogeneity of texture has influence

Effect of recrystallization on properties Hardness

Tensile strength Resistivity

Hardness: strongly reduced fro increasing rex Tensile strength: more pronounced reduction than for recovery resistivity: Back to pre-cold worked level due to annealing out of vacancies and intersticials

Main difference between

RECRYSTALLIZATION and

RECOVERY

Recovery: - partial restoration of properties - microstructure contains rearranged disloc. - grain structure related to deformed grains - some reduction in mechanical strength

compared to deformed metal Recrystallization - complete restoration of properties - microstructure = strain free - new grains formed, no resemblance to

deformed microstructure - material much softer after rex.

Defects in pure elements

Schottky: Atom moves from volume to surface Anti-Schottky: Atom moves from surface to volume Frenkel: Atom moves from volume to volume 0D: point defects 1D: dislocations 2D: interfaces, staking defaults 3D: new phases

difference constitutional vs. thermal vacancies

thermal vacancies: vacancies due to thermal equilibrium in material constitutional vacancies: alloying partners, present in high concentrations

Positioning of point defect in metals

positioning of vacancies self interstitials: symmetry of

octahedral/tetrahedral gaps in bcc, fcc crowdion definition

splits and their location in fcc, bcc

- vacancies not always at min. energy position:

must leave them to move. - vacancies can minimize energy if in pairs Self interstitials - octahedral & tetrahedral: in fcc: all same distance to interst.(cubic symmetry) in bcc: not all length the same, distorted lattice (tetrahedral symmetry) - crowdion: atom squeeze in close packed plane - splits: 2 atoms slightly displaced, fcc: in <100>, bcc: in <110>

Point defects in ion crystals

description of Anion/Kation vacancy detection of vacancy

fist detection due to…

Main name from colour (light abs.), a vacancy causes in ion. first detection of vacancies. Anion vacancy: F+

Kation vacancy: F-

Anion-vacancy+trapped electron: F Anion and Kation lattice can be treated separately Vacancy can be detected with xray

Source of vacancy: Thermal disorder

extensive quantities intensive quantities

extensive quantities: entropy S, volume V, nr. particles N intensive quantities: Temperature T, pressure P, chem potential µ (zero at equilibrium)

𝑐! = exp𝑠!𝑅

exp −ℎ!𝑅𝑇

ℎ!𝐴!

= 9𝑘!𝑇! = 1𝑒𝑉

𝑠!𝐴!

= 2𝑘!;            𝑐! 𝑇 = 𝑇! = 10!!

Methods to measure thermal vacancies I

Differential dilatometry

- direct determination of vacancy concentration - comparison of microscopic and macroscopic expansion: 𝑐! = 3(∆𝐿

𝐿− ∆𝑎

𝑎)

∆!!

= lattice parameter increase, xray at large angles ∆!!

= macroscopic expansion, laser interferometer both dominated by thermal expiation due to anharmonicity, But atoms relax around vacancies = lower a. Visible for T>0.8 Tm

Methods to measure thermal vacancies I

Differential dilatometry Problems

- only single vacancies are considered, but if curves are curved too much, esp. at elevated temp, we must consider double vacancies - high control over temperature and length required - measurements only visible at high temperature

Methods to measure thermal vacancies II

positron annihilation

just measurement of enthalpy of formation 1. source of positron: Na22 ->

Ne22+e++neutrino+gamma strahlung (START)

2. Thermalization: positron is slowed down in material within 20ps = 1mm

3. Interaction: a. positron hides in vacancy= positive spot b. Interaction with conduction electron =

STOP radiation of 2x511keV Doppler Broadening and Angular correlation

Methods to measure thermal vacancies II

positron annihilation

From aver. time (start - stop) to vacancy conc.: General assumption: - no detrapping of trapped positrons - trapping rate independent of Temperature - positrons can be annihilated in free or in

trapped state

           𝜇!𝑐! =𝜏 − 𝜏!

𝜏! − 𝜏

We assume that trapping is linear combination of trapping rate in free state and trapping in bound state. Only hL (not sL), for T~0.6Tm

Methods to measure thermal vacancies II

positron annihilation Problems

- Extrapolation at high & low temperatures to get 𝜏!  &  𝜏!.

- - thermal expansion, which is observed at low

temp. is neglected later - - detrapping and temp. dependency of trapping

rate is neglegted - - Only hL (not sL) - T~0.6Tm

Methods to measure thermal vacancies III

specific heat

General: Formation of point defects requires energy = increased enthalpy= increased heat capacity.

𝑐! = 𝑐! + 𝑐![𝛿ℎ!𝛿𝑇

+ℎ𝐿2

𝑅𝑇2  ]

Methods to measure thermal vacancies III

specific heat

Problems

- Approximation of single vacancies

- Extrapolation of c0 with constant slope for higher temperatures

- no determination of sL

Ficks law

Fick’s first law:

𝑗 = −𝐷𝛿𝑐𝛿𝑥+< 𝑣 >∗ 𝑐

Fick’s second law:

𝛿𝑐𝛿𝑡=𝛿𝛿𝑥(𝐷

𝛿𝑐𝛿𝑥) +

𝛿𝛿𝑥(< 𝑣 >∗ 𝑐)

drift, diffusion

Description of diffusion:

Case I: layer on substrate Case II: sandwich geometry

Case III: two different substrates

FOR ALL: D not depending on x!

𝑐 𝑥. 𝑡 =𝑄

𝐴 𝜋𝐷𝑡exp  (−

𝑥!

4𝐷𝑡  )

- Case I: A=1 - Case II: A=2

- Case III: For analysis, cut in thin layers, we

get gauss distribution as for I&II for every layer, superposition yields:

𝑐 𝑥, 𝑡 − 𝑐!𝑐! − 𝑐!

=12(1 − erf

𝑥2 𝐷𝑡

)

Case IV: Matano analysis

= introduction of D(x), which is now depending on x!! - introduction of Boltzmann Trafo: 𝜆 = !

!

- we solve Ficks second law with Boltzmann Transformation and Bondary condition of Matano plane: 𝑥𝑑𝑐 = 0!!

!! (set x=0)

𝐷 𝑐 = −12𝑡

𝑥𝑑𝑐!!!𝑑𝑐

𝑑𝑥

Random walks most important averages

Description of Diffusion with random walk.

Generally wrong: only true if there is no

correlation between two jumps, which does not hold for diffusion!

1. <R> = 0 2. <R2> = N*r2

With Brownian motion we can describe Ficks law assuming small steps (Taylor expansion)

Additionally we multiply steps with distribution function, which describes the probability for a step with a certain projected length in x-direct.:

𝑐 𝑥, 𝑡 + 𝜏 = 𝑐 𝑥 − 𝑋, 𝑡 𝑊 𝑋, 𝜏!

𝑊 𝑋, 𝜏 =1

2 𝜋𝐷𝑡exp  (−

𝑥!

4𝐷𝜏)

Brownian motion with correlation

From Ficks law: D=<x2>/(2  𝜏) and <x>=  𝜏 Γ!  𝑥! And introduction of jump frequency Γ!   𝐷 = !

!Γ!  𝑥!

Diffusion coefficients for basic structures: bcc & fcc: 𝐷! = Γ!𝑎! hdp: 𝐷! = 𝐷! =

!!

!(3Γ! + Γ!), 𝐷! =

!!Γ!𝑐!

For ideal hdp (c/a=sqrt(8/3): Dx=Dy=Dz

Mechanisms of Diffusion (6 Types)

1. Direct exchange (highly improbable), Atoms switch place directly with neighbour

2. Ring mechanism: Several atoms switch places in a turn, requires high coordination

3. interstitial mechanism: atoms move from one interstitial site to the next.

4. vacancy mechanism: Jump from normal site in a vacancy. High correlation

5. intersticalcy mechanism: Atom in interstitial pushes normal atom in interstitial

6. crowdion: as 5, relaxed over more atoms.

Jump frequency Γ!

Jump frequency depends on:

1. barrier height of enthalpy of the jump 2. geometry 3. fibration frequency

Γ! = 𝛽𝜈exp  (−𝑔!𝑅𝑇  )

Influence of Correlation f I D*=f D = Diffusion coefficient of tracer

Depending on dominating diffusion mechanism, we assume more/ less correlation: vacancy (high), interstitalcy (medium) Described by correlation factor:

𝒇 =< 𝒙𝟐 >

< 𝒙𝟐 >𝒓𝒂𝒏𝒅𝒐𝒎

for vacancy mech: 𝑓 ≈ 1 − 2 𝑧 for interst. mech: 𝑓 = 1−< 𝑐𝑜𝑠𝜃 >

Influence of Correlation f II

self diffusion Built up of diffusion coefficient

D=D(random walk)*energy barrier*entropic requirement. energy barrier and entropic barrier: each for jump and for vacancy formation proposes straight lines. Deviations occur due to …divacancies in fcc metals …2 intersecting lines = phase transformation f can be determined by the Isotope effect (measuring diffusion of both alloys gives one f)

Influence of Correlation f III

diffusion in alloys

difficult… five frequency model for fcc & infinite dilution for each combination of neighbourhood between alloying atom, normal atom and vacancy, we get a jumping frequency (5 total) from that we can determine a correlation factor

Kirkendall effect

main equations how it is measured

Note: as Matano plane ≠ Kirkendall plane explains that diffusion is driven by vacancy

mechanism and not by interstitial mechanism

1. two materials, bring them together 2. Interface is marked with something that does

not involve in diffusion 3. Measure, how the marker moves with

respect to macroscopic border consider two fluxes: diffusion relative to lattice, moving lattice planes. we get 2 equations: 𝑣 = !!!!!

!!!!!  !!!!"

   ;  𝐷 𝑐 = !!!!!!!!!!!!!!

v and D(c) are determined experiment. -> D

Kirkendall effect II

Methods to measure Diffusion coefficient D(c)

Tracer Diffusion

- Application of thin layer of diffusing element - Leave for diffusion at set temp for some time - extract concentration profile by: i) Rutherford backscattering ii) mechanical grinding >3 microns iii) sputtering > 1 nm variation of D with Temperature one can determine enthalpy of diffusion (proportional to -1/(4Dt)

Kirkendall effect III

Methods to measure Diffusion coefficient D(c)

Snoek effect

Measurement of jump frequencies without solving Ficks second equation Measurement of short distance diffusion in a KRZ structure with assymetric OCTAHEDRAL sites! 1. Application of strain: some octrahedral sites

become more preferable = motion into these ones.

2. Release of strain: Relaxation, measure time.

𝐷 =124  Γ𝑎

! =𝑎!

36  𝜏!

Kirkendall effect IV

Methods to measure Diffusion coefficient D(c)

Recovery process

Determination of vacancy processes by electrical conductivity. changes are very small to see change at all. stage I: Recombination of close Frenkel stage II: agglo and growth of interst. cluster stage III: vacancies get mobile, annihilation stage IV: growth of vacancy loops stage V: disassociation of all clusters change in resistivity proportional to vacancy concentration

Ideal solutions

- No interaction with neighbourhood - homogenuous solution

∆𝐻!"# = 0;              𝜇! =!"!!!

= 𝜇!! + 𝑘!𝑇𝑙𝑛(𝑐!) Real solutions

- Interaction only with nearest neighbour - Preferentially AA, BB or AB pairing

∆𝐻!"# ≠ 0;          

𝜇! =𝜕𝐺𝜕𝑁!

= 𝜇!! + 𝑘!𝑇𝑙𝑛(𝑎!)

Heterogenuous solutions What is it?

Basic equation for construction Driving force for phase separation

= solutions where > 1 phases (may) be present most important concept: In thermal equilibrium, the chem. potential of one component must be equal in all phases!

𝜇! =𝜕𝐺𝜕𝑐!

=𝜕𝐺𝜕𝑐!

=  𝜇!  

Decomposition mechanism

1. Mean field theory = assumption of a smooth connection between the G curves of two phases. should not be applied for metals! = Bergauf diffusion Binodale = Coexistance Curve Spinodale = Wendepunkt

2. Nucleation and Growth = unstable regions due to thermal fluctuations. Precipitates & parent liquid have always equilibrium compostion. = Bergab diffusion

Not distinguishable by morphology!

classical nucleation

Driving forces mathematical connection between chemical potential, critical radius, critical gibbs energy

and undercooling

general description:

∆𝐺 𝑅 = ∆𝑔!! + ∆𝑔!"4𝜋3𝑅! + 4𝜋𝑅!𝜎

∆𝑔!! = chemical energy ∆𝑔!"  = increased energy due to misfit

∆𝑔!!  ∝  ∆𝑇    𝑅∗ ∝ ∆𝑇!!

∆𝐺(𝑅∗) ∝ (∆𝑇)!!

Zeldovich factor

= width of potential barrier ca. kBT below the maxiumum

Series of Precipitation

1) GP I zones = precipitates in (001) plane = completely coherent

2) GP II = θ’’ = two GP zones enclosing some matrix atoms = completely coherent

3) θ’ = precipitates with atoms in (100) and (010) planes. (001) still coherent, rest incoherent

4) θ = completely incoherent, thermodynamically stable precipitates

Shape of precipitates

Depends on the elastic energy the system can afford. Higher elastic energy = less driving force for nucleation. plates = very low elastic energy spheres = very high elastic energy needle = average elastic energy cost.

Definition Ferroic Two main points

Example

1. Two or more orientational states;

identical, except for the direction of their orientation vector

2. Can be shifted from one to another by appling the conjugate field

Bsp: Ferromagnet, Energy for switching = Zeemann energy E=M*H Ferroelectric, Energy for switching = Electrostatic energy E=P*E (electric field)

Ferroics with zero M or P Possible explanations

1. Above Tc = curie Temperature of ferroic

ordering. Thermal agitation is larger than internal magnetizing field

2. Formation of domains = magnetized/ polarized regions of micro- to macroscopic size of different orientations, so that the total magnetization / polarization = 0

Main energies influencing domain formation, form and size

1. Magnetostatic energy: magnetic field in air generated due to connection of magnetic poles. Minimized by two antiparallel domains

2. Exchange energy: energy effort to keep two differently orientated domains next to each other. Minimized by introduction of closure domains.

3. Magnetocrystalline energy: energy difference between hard and easy axis. Minimized by magnetization in easy direction.

4. Magnetostrictive energy: domains elongate in field direction = elastic strain energy at domain boundarys. Minimization by reducing the size of domains of closure.

Origin of Magnetocrystalline Anisotropy

Spin- Orbit coupling. spin magnetic momentum: could align quite freely in the magnetic field. However, it is bound to the orbit magnetic momentum. orbit magnetic momentum: cannot move as freely. Depending on the orientation, orbitals overlap more or less = higher or lower energy states. Only if field is large enough, the orbit momentum can be aligned.

Size of Spin-Orbit coupling

Atomic mass Z4 Best atoms for large Spin-Orbit coupling have large anisotropy ( eg f-electrons, rare earth, La…L

u)

BCC hard, easy, intermediate direction

hard direction: body diagonal easy direction: along cube edge intermediate direction: face diagonal

Hysteresis at first magnetization Behaviour of Alignement

1. linear behaviour: Domain walls cannot

yet pass defects.

2. domain wall motions: Domain walls can pass defects, not completely reversible anymore

3. rotation of spins: Spins rotate even

against magnetostriction energy. Reversible as soon as field removed.

uniaxial anisotropy Behaviour of M-H curve for domains

perpendicular resp parallel to H

perpendicular: linear behaviour of the curve, no hysteresis, only rotation of spins. No domain wall motion parallel to H: rectangular hysteresis curve, no domain wall motion, spins suddenly rotate.

Defects in a material

consequence for hysteresis

Comparison remanence magnetization and saturation magnetization

application of such material

defect always associated with magnetostatic energy, reduced by inclusion of domain wall é defects = é hysteresis field but not rectangular, ie Mr < Ms inexpensive low end permanent magnet

Domain walls

which energies contribute to its size? comparison ferromagnets vs ferroelectrics

how is the wall turned?

exchange energy prefers wide walls (coulomb repulsion) magnetocrystalline anisotropy prefers narrow walls (only easy direction) Ferromagnets: rather wide walls, ca. 1 Micron Ferroelectrics: rather small walls, ca 2 unit cells wall is always turned such that no change in magnetization perpendicular to the wall does not change

Material properties to get wide walls

large spin orbit coupling = large magnetocrystalline anisotropy = large energy prefering narrow walls, as no hard direction should be present = more narrow walls

Desire towards single domain particles maximum size to get single domain

what determines domain size (saturation energy and domain wall energy)

size: not larger than domain wall, ie <1000A ê saturation energy = é critical size é domain wall energy = é critical size

shape anisotropy explanation, reason

how to maximize total anisotropy.

tends to make anisotropy lie along a long axis of the sample. reason: lower magnetostatic energy To get maximum anisotropy, align long axis of sample with easy direction of magnetocrystallinity.

Hysteresis curve for field:

- perpendicular to easy axis - parallel to easy axis

perpendicular: linear behaviour, no hysteresis parallel: large hysteresis, rectangular

Requirements for magnetic data storage media.

Requirements: - squared hysteresis (clear 1 or 0) - Mr: large to detect, small enough to prevent

magnetization of neighbouring domains - Hc: large to prevent flipping, small to allow

writing. - Tc: well above room temperature - cost: cheap= no rare earth - large anisotropy: as no rare earth: rectangular

shaped small particles - use of Fe, Co, Ni or alloys.

Werte für ∆𝑆, 𝑐!,𝐸,𝐺,𝑉𝑒𝑟𝑠𝑒𝑡𝑧𝑢𝑛𝑔𝑠𝑑𝑖𝑐ℎ𝑡𝑒  𝐵𝑢𝑟𝑔𝑒𝑟𝑠𝑣𝑒𝑘𝑡𝑜𝑟, 𝑙𝑎𝑡𝑒𝑛𝑡𝑒  𝑆𝑐ℎ𝑚𝑒𝑙𝑧

−𝑊ä𝑟𝑚𝑒,𝑅

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Kleinwinkelkorngrenze, Grosswinkelkg. kohärent, teilkohärent, inkohärent

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Dichte Schmelzpunkt

latent Schmelzwärme latent Sublimationswärme

Wärmekapazität E-Modul

Grenzflächenenergien.

Growth of nuclei

driven by… limited by

boundary conditions

proportionality of precipitated volume

Driven by: surrounding concentration limited by: rate of supply, rate of interface crossing boundary condition:

1. ejected atoms of matrix = absorbed atoms on precipitate.

2. total amount of B atoms stays constant precipitated vomume X(t):

𝑋 𝑡 ∝ 𝑡!/! Typical proportionallity for diffusion controlled

Growth of nuclei

for small volume: X(t) approach for large volume?

non precipitated volume: 1-X(t) for small X, this is a taylor expansion of exp(-x) non-precipitated volume:

= 1 − 𝑋 𝑡 = exp  (−(2𝑡3𝜏)!!  )

Ostwald Reipening

Origin of growth of larger grains

Difference in chemical potential: smaller precipitates= larger gas pressure = larger chemical potential= concentration gradient. LSW coarsening theory: boundary conditions

1. all phases are dilute solutions (no intermetallic phases)

2. small precipitated volumes 3. decomposition nearly completed

radius of grains: 𝑟 ∝ √𝑡

Field ion microscopy

evaporation mode

sample tip: 100nm field strength: 10 V/nm

1. Start signal: pulses on the general high field.

2. Ions evaporate from tip. 3. by detection of time of flight, we get

m/q of ion.

𝐸 =12𝑚𝑣! = 𝑞 ∗ 𝑈

Field ion microscopy

Image mode: general principal

typical characteristics of pictures

Helium in sample chamber is polarized close to tip. Electrons form He tunnels into tip = He+ and negative Ion in tip negatively charged Ion is rejected from tip. shows mainly areas of easy penetration

1. Rings: easy penetration of at lattice plane / tip boarder intersection

2. Black dots: tip surface is not smooth

small angle scattering

general principal, neuron treatement Assumption for scattering amplitude

difficulties

Neuron treatement:

1. Neurons are thermalized and made monochrome by monochromator

2. long tube to get parallel ray 3. scattering at large distance detector

Scattering amplitude: only difference between matrix and precipitates is interesting, rest is packed into single particle scattering function. Difficulties: particle size distribution!

Oder - disorder Transformation

Calculation of energy

𝐸 = 𝑐!𝑧2 𝑃!!𝜖!! + 𝑃!"𝜖!"

+ 𝑐!𝑧2 𝑃!!𝜖!! + 𝑃!"𝜖!"

Approach of parabola by setting P = cB Three cases: 1. down: ordering energy < pairing same energy 2. straigh: energy terms are equal 3. up: paring same energy > ordering energy.