Download - Rate Process 3rd Exam Lec1

8/3/2019 Rate Process 3rd Exam Lec1

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RATE PROCESSES3rd exam coverage


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reaction Rates application of reaction rates

to nucleation crystal growth

grain growth recrystallization precipitation sintering

oxidation solid state reactions role of kinetics in the development

of microstructure

Scope


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REFERENCES

1. Chemical Reaction Engineering. O. Levenspiel.Wiley (1972)

2. An Introduction to Transport Phenomena in MaterialsEngineering. D.R. Gaskell. Macmillan (1992).

3. Transport and Chemical Rate Phenomena, N.J. Themelis.Gordon and Breach (1995).

4. Transport Phenomena. A. Byron Bird., W.E. Stewart,E. N. Lightfoot. Wiley (1994).

5. Physical Chemistry of High Temperature Technology.

E.T. Turkdogan. Academic Press (1979).

6. Problems in Metallurgical Thermodynamics and Kinetics.G.S. Upadhyaya and R.K. Dube. Pergamon (1977).

7. Materials Science and Engineering An Introduction.William D. Callister, Jr. 5th edition (2000).


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REACTION RATES


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The rate of a reaction is the speed at which a reactionhappens. If a reaction has a low rate, that means themolecules combine at a slower speed than a reaction with ahigh rate. Some reactions take hundreds, maybe eventhousands of years while other can happen in less than one

second. The rate of reaction depends on the type of moleculesthat are combining.

RATES OF REACTIONS


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There is another big idea for rates of reactioncalled collision theory.

The collision theory says that the morecollisions in a system, the more likelycombinations of molecules will happen.

If there are a higher number of collisions in asystem, more combinations of molecules willoccur.

The reaction will go faster, and the rate of thatreaction will be higher.


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Reactions happen, no matter what.

Chemicals are always combining orbreaking down.

The reactions happen over and over but not

always at the same speed.

A few things affect the overall speed of thereaction and the number of collisions that

can occur.


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Concentration:

If there is more of a substance in a system,there is a greater chance that molecules willcollide and speed up the rate of the reaction.

If there is less of something, there will be fewercollisions and the reaction will probably happenat a slower speed.


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Temperature:

When you raise the temperature of a system,the molecules bounce around a lot more(because they have more energy).

When they bounce around more, they aremore likely to collide. That fact means theyare also more likely to combine.

When you lower the temperature, themolecules are slower and collide less. Thattemperature drop lowers the rate of thereaction.


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Pressure:

Pressure affects the rate of reaction, especiallywhen you look at gases.

When you increase the pressure, the moleculeshave less space in which they can move.

That greater concentration of moleculesincreases the number of collisions.

When you decrease the pressure, moleculesdon't hit each other as often.

The lower pressure decreases the rate ofreaction.


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APPLICATION OF REACTION

RATES IN NUCLEATION


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The Role of Nucleation and Growth inPhase Transformations

There are two distinct stages in a phasetransformation.

The first of these stages is referred to as

nucleation and the second stage isreferred to as growth.

The overall rate of a phase transformation

is determined by both the nucleation rateand the growth rate, since any given grainproduced by a phase transformation goesthrough sequential nucleation and growthstages.


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If two or more processes must occursimultaneously[i.e. in parallel] during areaction, the overall rate of reaction will bedetermined by theslowestof these.

However, if the processes must occuroneafter the other[i.e. serially], then theoverall rate of reaction will depend on therate ofallof the individual processes.


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For example, if you are building a house, thefoundation [and associated systems] must be laid

first and then the walls framed.

Thus, the time taken to the end of framing is thetime taken to lay the foundations,plus the timetaken to complete framing.

However, if the house is to be veneered withbrick, then bricks must be carried and mortarlaid at the same time.

If the bricklayer happens to be faster than themortar carrier, then the bricklayer must wait formore mortar to be brought up.


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Hence, the slowest member of the crew, in thiscase the mortar carrier, controls the rate atwhich the brick walls come up.

Try waiting for a contractor to finish a houseand you can observe this for yourself!

There is nothing intrinsically slower or fasterabout serial and parallel processes, instead thisdepends on the nature of the processesthemselves.

For example, in computers, a parallel printerport is faster than a serial communicationsport, but is slower than serial USB or[especially] Firewire ports.


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Nucleation

Consider the formation of a solid metal from itsliquid.

The production of a new grain begins when a

solid nucleus forms in the liquid.

Do not confuse this nucleus with the atomicnucleus - in the present context a nucleusmight contain up to a few hundred atoms.

If a solid nucleus forms in the liquid then a newsolid - liquid interface is produced.


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Like any other interface, this has a certain

interfacial energy per unit area of interface (SL)

and so this new interface adds energy to thesystem (this is represented mathematically by the

sign ofSL being positive).

Since a sphere has the lowest possible surfacearea to volume ratio, the nucleus minimizes itstotal interfacial energy by adopting a sphericalshape.

Interfacial energies do depend on temperature,but unless the temperature change is very large,

SL can be treated as a constant.


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The actual value ofSL is really very small.

For example around 250 mJ m-2 of solidto liquid interface could be regarded as

typical of a pure metal.

In comparison a single 100 W light bulbuses 100 J every second!

Nonetheless, it will be seen shortly thatinterfacial energies are crucially importantin nucleation.


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Producing a new solid will change the energy ofthe bulk material by a certain amount (GV) perunit volume of solid formed.

If GS and GL are respectively the energy of aunit volume of solid and that of the volume of

liquid that transformed to make a unit volumeof solid, then GS is added and GL is removedfor each unit volume of solid formed.

In other words, the value ofGV is given by:

GV = GS - GL


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What about the sign ofGV?

Unlike SL (which is only weakly sensitive to

temperature), GV depends crucially ontemperature.

Heres how:

At the equilibrium melting temperature (TM),by definition the liquid and solid phases musthave the same energy.

Thus GV is zero and there is nothermodynamic driving force fortransformation (either melting orsolidification).


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At a temperature T > TM

The energy of the liquid is lower than thatof the solid, so that GV is positive andthere is a thermodynamic driving forcefor melting.

At temperatures T < TM

The energy of the solid is lower than that

of the liquid, so that GV is negative andthere is a thermodynamic driving forcefor solidification.


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The undercooling (T) below the equilibriumtransformation temperature (i.e. TM) is definedas:

T = TM T

Thus, as T increases, GV becomesincreasingly negative, since the solid becomes

more and more stable than the liquid thefurther one goes below the equilibriumtransformation temperature.


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Now lets put the surface and bulk energiestogether to determine the overall change in

energy (GL-S) on production of a solid nucleus:

GL-S = SL x surface area of nucleus +GV x volume of nucleus

For a spherical nucleus of radius r, thisbecomes:

GL-S = 4r2SL + (4/3) r3GV


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First consider a case (Figure 1) in which T > 0 sothat GV is negative.

Remember that the SL term is invariably positivefor solidification and that mathematically r2initially goes up more quickly than - (r3) goesdown.

Thus, the value ofGL-S initially increases withincreasing r and only decreases again when acritical radius r* is exceeded.

Any nucleus that forms with a radius r less than r*is unstable and can reduce its energy by revertingto the liquid state.


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In contrast, any nucleus with a radius r greater

than r* is stable and can reduce its energy bycontinuing to grow.

It will also be noted that there is an activationbarrier of magnitude G * to assembling anucleus of radius r*.

Figure 1: GL-S as a function of r
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Figure 1. GL-S as a function of r
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The values ofG* and r* depend crucially onT. As has already been noted, when T = 0, the

GV term = 0 and so there is nothing to offset theSL term and hence both G * and r* are infinite(see figure 2).

Since it is clearly impossible to assemble anucleus of infinite size, or to expend an infiniteamount of energy doing so, solidification clearlycan not take place at the equilibrium meltingtemperature TM.

In contrast, as G increases, the GV termbecomes increasingly negative and so the valuesof both G * and r* fall.


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Figure 2: The effect ofT on G* and r*.
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What determines the value ofT atwhich stable nucleation actuallyoccurs (TN)?

A vital factor is the cooling rate (dT/dt).

When the cooling rate is low, there is more timefor assembly of a nucleus of a given size thanwhen the cooling rate is high.


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Consequently:

As dT/dt increases, the typical value of rdecreases.

To allow stable nucleation r > r* and so,as dT/dt increases, nucleation will only beable to occur if r* is decreased.

A decrease in r* requires an increase in T.

As dT/dt increases, TN increases.


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Thus, when the cooling rate is very

high, nucleation will take place atextremely high undercooling.

Of course, solidification usuallyinvolves the formation of manynuclei, not just one and so the rate of

formation of nuclei is also important,as follows:


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As soon as T exceeds TN stable nuclei will beformed.

The greater the value ofTN the larger thedriving force for nucleation (since Gvbecomes increasingly negative as T increases).

The greater the driving force for nucleation, thegreater the rate (dN/dt, if the number of nucleiis N) at which these will form.

The higher dT/dt is the greater the value ofTNand hence the higher dN/dt.

Thus, when a very high cooling rate is used, many nucleiare formed.


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Homogeneous versusHeterogeneous Nucleation

In the example considered in the last section,solid nuclei were formed at random locations inthe liquid.

This is called homogeneous nucleation.

In contrast, what would happen if nucleationoccurred preferentially at the solid mold walls of

a casting instead of homogeneously.

Nucleation at preferred sites is referred to asheterogeneous nucleation.


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What is gained by nucleating

heterogeneously, instead ofhomogeneously?

Homogeneous nucleation:

A new solid - liquid interface is created and this isnot offset by the removal of any existinginterfacial area.

The result is a large increase in interfacial energy.


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Heterogeneous nucleation:

By nucleating, for example, on the walls of themold, an area of interface between the mold andthe liquid is removed.

This is then replaced by a new solid to liquid

interface and a new solid to mold wall interface.

Removing some of the existing mold to liquidinterface helps to offset the interfacial energy

added by the new interfaces.

The result can be a much smaller increase ininterfacial energy than for homogeneousnucleation.


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How can the effectiveness of

heterogeneous nucleation bedetermined?

The best place to start is to consider the Young

equation for the balance of interfacial tensions forthe geometry shown in figure 3.

Figure 3: Derivation of the Young equation.
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Figure 3: Derivation of the Young equation.
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By resolving forces in the horizontal direction it canbe seen that:

ML = SM + SL cos

Where:

is known as the contact angle and the

terms represent the interfacial tensionsof the mold to liquid (ML), solid to mold (SM)and solid to liquid (SL) interfaces.


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Note:

Interfacial tensions have units of N m-1, whereas

the units of interfacial energy are J m-2. So longas consistent units are used [e.g. do not mix SIand non-SI units], the interfacial tension andinterfacial energy are numerically equal and the

same symbol g is used for both.

However, within a single calculation, interfacialtensions and energies should not be mixed. Forexample, one can describe the contents of a bottleof soda as still 20 percent full [the optimistsview!] or 80 percent empty [the pessimistsview!], but mixing these up when trying to workout how many drinks are left will give nonsense.


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The lower the value of the contact angle, the moreeffectively the solid nucleus wets the mold and

the more effective a heterogeneous nucleation sitethe mold will be.

Remember that cos (0)= 1 and cos (90) = 0 so for

best wetting (i.e. q tending to zero), cos shouldbe as high as possible.

In other words, the higher the value ofML and the

lower the value ofSM, the more effective themold will be as a heterogeneous nucleation site.


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Does using the Young equation to explainheterogeneous nucleation make sense?

Yes, because to be effective as a heterogeneousnucleation site:

the amount of interfacial energy added by the

presence of the nucleus should be as small aspossible;

the amount of interfacial energy removed bythe presence of the nucleus should be as large

as possible;

so that, on balance, the least possible amount oftotal interfacial energy is added.


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Thus, the most potent heterogeneousnucleation sites for new solid are those that havea high interfacial energy with the liquid and a lowinterfacial energy with the solid nucleus.

Looking at the Young equation, these sameconditions will result in efficient wetting of the

heterogeneous nucleation site by the new solid.

(The reason that the terms wet and wettingare used in the context of a solid nucleus forming

on a solid surface is that contact angles were firstconsidered for liquid droplets wetting solidsurfaces and nucleation theory is much morerecent.)


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The effect of heterogeneous nucleation onthe activation barrier to nucleation is

represented by a function called theshape factor, which has the symbol

S().

The exact derivation of S() depends onthe shape of the nucleus, but as q

decreases so does S().

When is low, S()


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However:

G*HET

= G*HOM

S()

where HET and HOM refer to heterogeneous andhomogeneous nucleation.

The availability of potent heterogeneous nucleation

sites, i.e. sites that result in a low S(), greatlyreduces the size of the activation barrier tonucleation.

Thus, the value ofTN for heterogeneousnucleation is often much smaller than forhomogeneous nucleation.


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Hence,

in most cases, nucleation occursheterogeneously, rather thanhomogeneously.

The example given here has focused onsolidification.

However, the same basic theory is alsoapplicable to solid-state phasetransformations.


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In the case of solid-state phase transformations,nucleation usually occurs heterogeneously, often on

grain boundaries (although other heterogeneousnucleation sites are also important).

Homogeneous nucleation is only observed in caseswhere the interfacial energy between a second phaseand the matrix in which this precipitates is very small.

For this to happen, the precipitate and matrix need tohave similar crystal structures, lattice parameters and

suitable chemistry. In cases where the interfacialenergy between the precipitate and matrix is verysmall, homogeneous nucleation is possible becauseG*HOM is low, even at relatively small T.


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Growth

Once stable nucleation has been

achieved, the growth stage of thephase transformation begins.

As the name of this stage implies,during growth the stable nucleusproceeds to grow into a grain.


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For growth to occur, two thingsmust happen:

The interface between the productand parent phases (the phase

boundary) must be mobile (i.e.able to move), otherwise the newphase cant grow.


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If, according to the phase diagram, there is adifference in composition between the twophases, then it will be necessary to transfer

solute from the parent to the product phase (orvice-versa, depending on the compositions of thetwo phases).

For example, if the carbon-rich phase Fe3C is toprecipitate in a matrix of-Fe, then carbon must

be transferred from the -Fe to the Fe3C.

The mechanism by which the composition of theparent and product phases is adjusted involves thediffusion of solute.


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Since motion of the phase boundary and diffusionmust occur at the same time, whichever of these

two processes is slowest will control the overallrate of growth.

Cases where mobility of the interface is the rate

limiting step are referred to as interfacecontrolled, whilst those in which diffusion is therate limiting step are known as diffusioncontrolled.

Most, but not all, phase transformations arediffusion controlled.


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As an analogy, think about what controlsthe rate of someone walking across a

floor.

Normally, the rate controlling step is therate at which one can walk - think of this

as being like diffusion control.

If, however, something very sticky is split

on the floor, then the rate controlling stepbecomes detaching ones feet from thefloor - think of this as being like interfacecontrol.


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The Nature of Diffusion

Diffusion involves the random-walk of atoms.

In this type of motion (which is approximated,but not usually achieved, by persons whom have

consumed an excess of certain ethanol-bearingbeverages) each atom moves by a series ofrandom jumps, independently of other atoms.

The effect of this random walk is that anygradient in concentration is gradually removed.


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In other words, diffusion tends tohomogenize.

To verify this yourself, try experimentingwith putting drops of food coloring intowater and watching what happens to the

color over time (also try changing thetemperature of the water and observethe effect this has on the rate ofhomogenization).


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How is homogenization possible ifdiffusion is a random process?

Consider a case when NL atoms arepresent on the left hand side of aspecimen and NR atoms are present on

the right hand side.

In a random walk process, the probabilityof any given atom jumping from left toright (PLR) is the same as that from rightto left (PRL).


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The behavior of any individual atom can not bepredicted, but the overall number of jumps from

left to right (NLR) and right to left (NRL) in time tcan be predicted since:

NLR = NL PLRNRL = NR PRL

The net motion of atoms from left to right (NNET)would then be:

NNET = NLR- NRL = NL PLR - NRPRL= P (NL - NR)

where P = PLR = PRL.


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Thus, so long as there is a gradient inconcentration, the net flow of atoms will be

from the concentrated to the lessconcentrated side (i.e. diffusion tends tohomogenize).

In other words, diffusion occurs down aconcentration gradient.

How can this be representedmathematically?

Ficks first law of diffusion states


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Ficks first law of diffusion states:

J = -D dc/dx

where:

J = flux (i.e. the number of atoms passing through

1 m2

of a reference plane per second),

c = concentration and

x = distance (so that dc/dx is the concentrationgradient).

The D term is known as the diffusion coefficient ordiffusivity and has units of m2 s-1.


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The flux is a measure of how fast material isactually transported by diffusion.

The (dc/dx) term and the minus sign reflectdiffusion taking place down a concentrationgradient.

The diffusion coefficient is a measure of howrapidly a species is able to diffuse.

Consider an analogy. Feeling fed up with the

crowds in the city and wanting to get away frompeople provides the motivation to visit thecountryside where there are fewer people (theconcentration gradient).

Having a car provides a faster means of


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Having a car provides a faster means ofgetting there than walking (thediffusivity).

Thus, if a lot of fed-up city people havecars then there will be many peoplemoving through the city limits in a givenperiod of time (the flux).


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What controls the value of D?

The answer to this is two things, firstly the soluteand solvent involved (i.e. D is material specific)and secondly D depends on temperature (i.e.diffusion is thermally activated).

Mathematically, the effect of temperature andmaterial on diffusion is represented by theArrhenius relationship:

D = D0 exp [-Q / (RT)]


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where:

D0 is a material-specific constant,

R is the gas constant and T the temperature.

Q represents an activation energy.

t is not surprising to realize that increasing thetemperature makes diffusion faster, sinceincreasing the temperature would be expected toincrease the frequency with which atoms jump.

However, to really understand diffusion, it isnecessary to understand the role that thematerial plays in this process.


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Solid-State Diffusion

In a liquid or gas, the atoms (ormolecules, or ions as appropriate)are clearly free to move and this

allows diffusion to occur.

In the solid-state, however, it is

less clear that diffusion can occurand a mechanism is needed thatallows atoms to jump.


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At any temperature above absolute zero (0 K, i.e.- 273 C) some of the atomic sites in any crystalwill be vacant at equilibrium and the number ofvacant sites (vacancies) increases withtemperature by an Arhennius relationship.

This raises the possibility of diffusion occurringby atoms swapping places with vacant sites (seefigure 4).

In this way, both the solvent atoms (selfdiffusion) and substitutional solutes can diffuse.


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Figure 4: Vacancy - atom interchange
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The rate at which diffusion can occur will thendepend on the availability and mobility ofvacancies.

Hence the activation energy Q term in theArhennius relationship actually consists of:

Q = QF + QMwhere

QF and QM respectively represent the activation

energies for the creation and migration ofvacancies.

Q is a material (i.e. solute and solvent) specificconstant with units of kJ mol-1.

B th th f ti d i ti f i


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Both the formation and migration of vacanciesbecome easier as the temperature is increasedbecause:

Vacancies are created by atoms being shakenloose at interfaces such as grain boundaries,leaving a vacant site (see figure 5).

The shaking comes from thermally-inducedvibration of the atoms making up the lattice of thecrystal.

Hence, on-heating, the number of vacanciesincreases. Conversely, on-cooling, atoms presentat boundaries tend to drop back into the vacantsites and the number of vacancies decreases.


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Figure 5: Creation of vacancies.

1 2 3

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When a vacancy and an atom swap

places, the atom is temporarily out ofits equilibrium position and has tosqueeze past neighboring atoms (figure 6).

This becomes easier as the temperatureincreases, because thermally-inducedlattice vibration dilates the lattice(i.e. lattice parameters increase withtemperature).


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Figure 6 : Activation barrier to vacancy atom interchange.
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Consequently, whereas a given metal (saycopper) might have only one vacancy -

atom interchange expected in billions ofyears at room temperature, millions ofthese interchanges will take place persecond just below the metals melting

temperature (1,083 C for copper).

Hence, there is a very strong positive effectof the temperature (T) term on the value ofthe diffusivity D.


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As has already been noted,the activation energy

(Q) is a material specific constant and the valueof this reflects the relative ease with whichvacancies can be created and move.

The more strongly bound the material, the harder

will be both of these two processes.

Hence, Q tends to increase with the meltingtemperature of the material, since a high melting

temperature is indicative of strong bonding.

Th i


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Thus, at a given temperature,tungsten will have a lower self

diffusivity than say chromium (eventhough these are both BCC metals).

Interestingly enough, at atemperature just below theirrespective melting point, differentmetals with the same crystalstructure has about the samediffusivity.


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Thus far, we have considered self diffusionand the diffusion of substitutional solutes,

but not the diffusion of interstitials.

In the case of interstitial solutes, mostinterstitial sites are empty and so

vacancies do not need to be created.

Thus, the activation energy for interstitialdiffusion is governed by the strain imposed

on the lattice of the solvent as the soluteatoms jump from interstitial site tointerstitial site.


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If the interstitial is very small, say atomichydrogen (i.e. H and not molecular hydrogen H2)in iron then this strain will be quite small and theactivation energy for diffusion will be very low.

Even with larger interstitial solutes, like carbon iniron, the activation energy for interstitial diffusion

will be much lower than for self diffusion.

Consequently, at any given temperature, thediffusivity of interstitial species will be much

higher than for substitutional solutes, or selfdiffusion.

Th Eff t f N l ti d G th


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The Effect of Nucleation and Growthon Microstructural Development

The nature of the nucleation and growth processhas a profound effect on microstructuraldevelopment.

For now, two key effects of nucleation andgrowth on microstructural development will bediscussed:


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When are diffusional phasetransformations fastest?

In other words, how much undercooling will givethe fastest rate of transformation?

As TN increases, nucleation occurs faster

However, an increase in TN implies that theactual temperature is lower and hence diffusion

controlled growth becomes slower.

Balancing these two factors results in fastesttransformation at intermediate values ofTN.


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How can the grain size be controlled?

Increasing the cooling rate will increase

TN.

This will make nucleation of grains fasterand growth of grains slower.

Consequently as dT/dt increases, the meangrain size decreases.

I


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In summary:

Phase transformations inceramic materials are fairlysimilar to those in metals. The

behavior of polymers is quitedifferent to that of metals, butsome of the basic concepts

discussed here are stillapplicable.