Chapter 8 Solidification

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    Chapter 8 Solidification

    8.1 The liquid state

    Thermal energy per mole, RTm heat of melting per

    mole, Hm(Richards rule) Hm RTmAt Tm : liquid and solid in thermal equilibrium (coexist) for

    constant pressure

    Gibb free energy:

    Gliquid = GsolidHliquid Tm.Sliquid = HsolidTm.SsolidHliquid Hsolid = Tm(Ssolid Sliquid)

    Hm = Tm.Sm

    Hm - heat of melting, table 8.1

    Sm - the melting entropy

    Most metals crystallize in close-packed crystal structure(fcc or hcp)

    Crystalline state has higher packing density and lowvolume than those of the liquid state, fig. 8.1.

    Solid Liquid

    Heat depends on binding force between atomswhich depends on metals

    But for non close-packed crystals:

    State: Crystal Melt

    Volume: high low

    Ex: Si dc (diamond cubic) open structure low densityTable 8.2

    Atomic arrangement of atoms in a melt

    - is not entirely random

    - locally dense arrangement but not as densely packedas in a close-packed crystalline solid, fig 8.2.

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    Fig 8.2 a - Liquid Zn at 460oC (Tm = 419.4oC)

    -Broken line completely random atomic distribution as ina gas

    - Solid line specific r(spacing) high intensity

    Fig 8.2 b number of atoms as a function of spacingcalculated from fig. 8.2 a.

    -Broken line random distribution

    -Solid line spacing calculated

    Distance between 2 atoms for small value of r

    -In liquid state similar to solid state

    -Have the same number of next neighbor atoms, fig. 8.2 b.

    Conclusion:

    -The melt is more similar to crystalline state than to acompletely random state as a gas.

    - liquid state short range order

    - solid state long range order

    The concept

    -The crystal structure break down during melting

    - There is thermal fluctuations cause decomposition and

    formation of an ordered dense atomic arrangement- Dynamic formation and destruction of locally orderedstructures affect the nucleation of the solid state uponcooling below melting temperature.

    The vertical lines in fig. 8.2 b are thespacing of atom in crystalline zinc.

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    8.2 Nucleation of the solid state

    T < Tm the free energy of crystalline state is the lowestT > Tm - the free energy exceeds the free energy of the

    liquid stateT = Tm - the free energy of liquid = solid or in equilibrium

    The free energy of crystalline solid and liquid decreasewith decreasing temperature, fig. 8.3.

    Fig. 8.3- solid lines free energy in equilibrium- if the crystal is superheated or the melt is supercooled,

    there is a driving force (free energy /unit volume) tochange the state.

    the driving force (free energy/unit volume)gu = (Gliquid Gsolid)/V

    The solid state- is not obtained spontaneously- necessary to have a certain supercooling- a small volume of crystalline has to form through

    thermal fluctuation.- thermal fluctuations occur in liquid state due to motion

    of the atoms.T > Tm nucleus or embryo is unstable, will decomposeT < Tm - nucleus becomes stable only if it exceeds a

    critical size.

    Assume: nucleus is a spherical shape with radius r.

    The size of nucleus affect by- Solid-liquid surface energy is always posit ive

    - nucleus is very small at first growfree energy increases with increasing size

    Formation of the nucleus the change of free energy ofthe system (GN) is the sum of 2 terms.

    1. Energy from formation of solid :the volume energy = 4/3 r3 .(- gu)

    2. The surface energy = 4 r2 is the solid-liquid surface energy per unit area

    GN = - 4/3 r3 .gu + 4 r2 8.1

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    T > Tm - the change of volume free energy gu 0

    - therefore, GN is always positive

    - any embryonic nucleus decomposes with a

    gain of free energy, fig 8.4.

    T < Tm - the free energy of nucleus decreases only if

    the nucleus exceeds the size r ro

    At r = ro : the work of nucleation GN , attain a maximum

    fig. 8.4 b

    r can be calculated from eq. 8.1 by differentiation

    d (GN )/dr = 0

    r = 2 / gu 8.2

    Fig. 8.4 b: r = ro

    G(ro

    ) = 1/3 Ao

    8.3

    Ao is the surface area of the critical nucleus

    From eq. 8.3: For formation of a viable nucleus,

    the required gained of volume free energydoes not have to compensate fully the surface energy ofthe nucleus but only 1/3 of it.

    the volume energy compensate only 1/3 of

    surface energy of nucleus becausethe free energy of the system will decrease

    by the growth of a nucleus of size r > ro even throughthe absolute of the free energy is larger than in a systemwithout nucleus.

    Nucleation occurs through thermal fluctuations - the frequency of nucleus formation per unit volume

    and unit time ~

    exp (-Go/RT) 8.4

    lim Go/RT = TTm, T0

    T Tm : gu 0, Go becomes infinitely large

    T 0 : gu remain finite, Go also a finite value but

    but Go/RT

    ~

    exp (-Go/RT) (0) = 0

    T = 0 : = 0

    0

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    depends on temperature

    gu depends on temperature

    Go is a function of temperature (via gu)

    Therefore, small changes (in temperature) insupercooling results in large changes of nucleation rate,

    fig. 8.7.

    Nucleation rate obtains from eq. 8.4 is much smallerthan that observed in reality because the nucleationoccur inhomogeneously within the volume, fig. 8.8.

    Nucleation occur at the pre-existing surfaces, example

    - on a mold wall

    - on particles in the melt

    Part of the nucleus surface is provided by the alreadyexisting surface of the particle which reduces the workof nucleation.

    For heterogeneous nucleation on a flat surface, thework of nucleation

    Ghet = f.Go 8.5

    f = (2 + cos) (1 cos)2 8.6

    is the contact angle, fig. 8.9.

    0 0 f 1

    strongly depend on G

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    Therefore, heterogeneous nucleation rate is much largerthan the homogeneous nucleation rate (especially atsmall supercooling)

    In commercial ingot making process: insoluble particlesare added to the melt prior to solidification.

    ex, TiB2 particle added to aluminum- increase the number of nuclei

    - produce a fine grain size.

    Nucleation rate

    - depends on superheating of the melt prior tosolidification

    - particles in the melt which act as heterogeneousnucleation can dissolve with increasing temperature

    - depends on the alloy composition, fig. 8.10.

    In commercial alloy

    -Supercooling occurs only a few degrees becausenucleation occurs by heterogeneous nucleation.

    - homogeneous nucleation hardly occurs because thereare always have high contaminating particles to induceheterogeneous nucleation, fig. 8.11.

    Homogeneous nucleation occur only in the experiment ofa small droplet of the melt

    - a small particle-free droplets crystallized only atlarge supercooling .

    Pure metals can be substantially undercooled by amountof 15% of the absolute temperature, table 8.3.

    A large supercooling can be obtained in organic materialsbecause there more complicated molecularstructure substantially impede crystallization

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    The shape of a crystal during solidification is usually notthe equilibrium shape.

    A spherical shape is obtained only if the crystalgrows equally fast in all direction.

    A polyhedron is obtained where the surface is

    composed of the most slowly growing crystallographicplanes because the fast growing planes disappear in thecourse of growth, fig. 8.14.

    A dendritic crystal forms because of instabilitiesin the shape caused by transport processes.

    8.3.2 Atomistics of crystal growth

    Kossel and Stranski studied the atomistic processesof crystal growth from the vapor phase: their theory:

    -Atoms adsorbed on a flat surface, generate a nucleusthen nucleus grows by attachment of further atoms untila complete new layer forms.

    -For a further growth, the generation of a new nucleus isrequired and so on.

    - during deposition of atoms from vapor phase onto asolid surface a variety of atomic configurations on thesurface can be established, fig 8.15.

    - Formation of a nucleus is much slower than the growthof a layer. Therefore, nucleation is the rate controllingprocess in the model.

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    In experiment

    -The growth rate increase with increasing supercoolingat rate much faster than predicted by the model.

    - the reason for this : the existence defects (screwdislocation) act as a nucleus, Nucleation is not needed,cause high growth rate.

    8.3.3 Crystal growth in the melt

    8.3.3.1 Solidification of pure metal

    -The shape of grains in a solidified metal is determinedby heat flux distribution during solidification. The heatremoved either through the solid or through the melt.

    - At solidification front, a discontinuity in the temperaturegradient occurs because of the latent heat, fig 8.18.

    - A displacement of the solidification front by dx in a timeinterval dt, generate heat flux, hS, per unit of crosssection and time.

    hS = dx/dt = hSv

    hS - the heat of solidification per unit volume

    -If the latent heat flow through the solid

    Total heat removed = the sum of heat flux from liquid +

    the latent heat of solidification

    heat flux density, j = (dT/dx)

    L and C - thermal conductivities in the melt and crystal

    The balance of heat flux:

    C (dT/dx)C - L (dT/dx)L = hS.v 8.8

    If the heat flow occurs through the crystal (fig.8.18a) then the temperature gradient in the crystal must belarger than that in the melt.

    If the solidification front grow faster and extends into themelt then the extended region will dissolve. Therefore,

    solidification front remains flat and grow in stable fashion,equiaxed grains develop.

    If the heat flux occurs through the melt (fig. 8.18b)

    The melt is strongly supercooled during solidification.

    -Any irregularity develops at the solidification front will beextend into a volume of undercooled melt where it cangrow faster.

    - slender crystal form branch in perpendicular

    directions the solidification front does not move in astable way

    - dendritics structure obtained.

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    During casting into a cold mold

    -At the mold wall, the melt is strongly supercooled(broken line in fig 8.21)

    - When solidification starts and proceeds away from themold wall, temperature of crystal (away from the moldwall) increases until reaches Tm. Then temperature ofthe solidification front decrease to minimum.

    - the dendritic formation

    Fig. 8.23, a binary alloy with complete solubility

    Assume heat remove through crystal

    Right hand side

    solid line temperature profile

    broken line temperature of solidification front

    dotted line the liquidus temperature of the melt that

    changes with compositions

    Left hand side

    solid line concentration profile in crystal and melt at

    the solidification front

    broken line the composition of crystal and melt at

    the solidification front that moves through

    left to right

    Diffusion in the crystal and melt is fast, fig. 8.23a.

    - after solidification, the solid has a homogeneouscomposition, co = the initial melt.

    Diffusion in crystal is low but in the melt is fast,

    fig 8.23b.

    -The composition homogeneous in the melt but not in thesolid.

    - the melt becomes enriched with solute atoms- composition of the melt can be higher than c2- after solidification, there remains a residual concentrationgradient in crystal.

    8.3.3.2 Solidification of alloys

    - Alloys solidify over a temperature range(between liquidus and solidus temperature)

    - During solidif ication melt and crystal coexist inequilibrium but have different compositions.

    - diffusion will be taken into account in thesolidification of alloys.

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    Diffusion proceeds slowly both in the crystal and in

    the melt, fig 8.23c.

    - The compositions in the melt and solid are not

    homogeneous

    - The solute atoms are ejected from the crystal at the

    solidification front and remain in the melt close tothe solidification front, in the distance of .

    - The composition further away from the solidification frontremain unchanged or at co

    - The maximum concentration of the melt at thesolidification front can only be c2.

    - Crystal solidified with composition co and thecomposition of the melt near the solidification frontis c2. the composition of the melt changes rapidly

    in the vicinity of the solidification front from c2 to coin distance of

    - therefore, temperature increases from T2 to T1 in thesame distance of

    - If the temperature increase is larger than the actualtemperature gradient in the melt. The temperaturein front of the solidification front in the melt will belower than the liquidus temperature of the meltwith conc. co called constitutional supercooling

    because of the supercooling is associated withlocal constitution. Dendritics are formed duringsolidification although there is no thermalsupercooling of the melt.

    - The dendrite in alloy is caused by constitutionsupercooling.

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    8.3.3.3 Solidification of eutectic alloys

    - Alloys with eutectic composition solidify at a fixedtemperature, forming 2 phases.

    - exchange of atoms by diffusion

    - time is limited the region of composition change isalso limited, result in lamellar microstructure

    - If the alloys composition is different from eutectic, theprimary phase is dendritic because of constitutionalsupercooling

    - the melt of the eutectic composition solidifies lamellar structure

    - lamellar spacing, l , can be determined by cooling rate,R

    l2 . R = constant 8.9

    high cooling rate small lamellae spacing

    - Direction of solidification cause lamellae directioncorresponds to a fiber or lamellar reinforced composite,fig. 8.26

    8.4 Microstructure of a cast ingot

    The typical microstructure of cast ingot consist of 3zones, fig 8.27

    a. The fine-grained chill zone

    b. The columnar zone

    c. The equiaxed zone

    Solidificaiton

    - Heterogeneous nucleation at the mold wall growth toward the center of the mold

    - the growth rate depends on its crystallographicorientation. The surviving grains exhibit a preferredcrystallographic direction of the grains long axis.

    - During the solidification of the 2 zones, impurities

    are ejected into the remaining melt and served asnuclei. The fine equiaxed grains are obtained atthe center of the ingot.

    - If the metals is pure, the equiaxed zonedisappears.

    - the width of the individual casting zone dependson solidification condition.

    - If the temperature of the melt is high, theimpurities particles dissolve and the cooling rate is

    low columnar microstructure forms.- Lower casting temperature and mold temperature

    induces a higher nucleation rate and equiaxedzone at the center increases.

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    8.5 Solidification defects

    Most metal

    -Have larger volume in liquid state than in the solid state

    - hole or pipe can be formed at the center of the castingingot after liquid metal solidify

    - or macroscopic shrinkage hole, microscopic cavitiescan form by contraction.

    A liquid metal can dissolve a much larger amount ofgas than the crystalline solid (fig. 8.32)

    during solidification gas bubbles form

    - rise in liquid metal cause mixing of the melt

    - trapped in the ingot pores gas porosity

    Segregation

    -Gravity segregation results from the solidificationcrystals have a density different from the density of theliquid. The less dense crystals are lighter and will be atthe top of the ingot. The heavier crystals will be at thebottom.

    - Macrosegregation caused by impurities segregate atvarious locations of the cast ingot.

    cannot be removed by heat treatment

    prevent by stirring

    -Microsegregation is concentration gradient in solidsolution crystals.

    Composition differences occur during solidification

    Can be removed by subsequent homogenization heattreatment

    8.6 Rapid solidification of metals and alloys

    Solidification with high cooling rate

    -Grain size decreases with increasing cooling rate

    (grain size < 1 micron)

    -Supersaturated solid solution can occur

    - system with intermetallic compounds, metastable phaseswith simpler crystal structure can be obtained.- In some special system (Al-Mn) with high cooling ratescan yield quasicrystals

    Quasicrystals

    - Is obtained from a mixture of two different structureelements

    - does not generate a long range ordered atomic

    arrangements- has a five-fold rotation symmetry (called Penrosepatterns, fig. 8.35)

    - application

    some systems exhibit high hardness and limited

    wetability used for coating (frying pan)

    -If cooling rates are extremely high (105 106 K/s) alloysand pure metals will solidify as an amorphous structure metallic glasses.

    - To obtain a metallic glass by solidification, the melt mustbe rapidly cooled below the solidification temperature. Thealloy with low melting temperature has an advantage toproduce metallic glass because eutectic temperature isclose to its glass transition temperature.

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    - metallic glasses are unstable during annealing

    During temperature increase

    a structure relaxation occurs

    Atoms move to more stable arrangements

    Cause embrittlement

    At higher temperature: crystallization occurs. Theservice temperature of metallic glasses is limited bytheir devitrification.