Marder M.P. Condensed Matter Physics

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Alloys 103

and CuNi. All of these elements share an fee structure in the ground state, members

of the pair have very similar lattice constants (see Table 2.1), and the two metals

mix substitutionally into each other in any concentration. The Au-Cu phase dia-

gram provides an example in Figure 5.3(A). Such unlimited solubility is, however,

quite rare, and even elements with the same crystal structure and similar lattice

constants typically pass through an elaborate sequence of structures. For example,

Ag and Al share the fee structure and have lattice constants differing only by 1%,

yet Ag is soluble in Al up to only 0.2% at 200 °C.

Gold

Copper

Silver

Copper

(A)

10 20 30 40 50 60 70 80 90

Atomic Percent Copper (B)

0 20 30 40 50 60 70 80 90 100

Atomic Percent Copper

Figure 5.3. (A) Phase diagram of copper and gold, which form a perfect substitutional

alloy but have a number of superlattice structures at intermediate temperatures. (B) Phase

diagram of copper and silver, which have limited solubility and form a eutectic. [After

Hansen (1958).]

In general, beyond a critical concentration that depends upon temperature, the

atoms of alloys begin to array themselves into an immense variety of structures.

The term intermetallic compound applies whenever two metals form a definite

crystal structure at some definite concentration. The crystals listed in Tables 2.2,

2.4, 2.5, 2.6, and 2.7 provide many examples. When the concentrations of ele-

ments vary continuously between values at which intermetallic compounds form,

they form secondary alloys, which can be thought of as intermetallic compounds

with excess or deficit of some elements. In addition, there are two general types

of ordering that can occur. If unlike atoms find it favorable to be near one another,

they may choose to form superlattices, particularly near favorable concentrations.

If, on the other hand, unlike atoms find each others' company energetically unsat-

isfactory, there will be a tendency toward phase separation.

5.3.3 Superlattices

The phase diagram of Figure 5.3(A) shows that gold mixes substitutionally into

a primary alloy of copper at all concentrations. The arrangement of the gold

with respect to the copper is not, however, always random. At temperatures be-

104

Chapter 5. Beyond Crystals

low 400 °C, X-ray diffraction off copper-gold mixtures, with three parts copper

to one part gold, shows a new set of diffraction peaks not present above 400 °C.

Furthermore, the strength of these peaks depends upon the way the mixture is pre-

pared. Bragg and Williams (1934) report that lowering of temperature from above

400 °C to 270 °C in a few seconds gives peaks no different from those present

above 400 °C—the mixture has been quenched. By cooling the solid solution in-

stead over a period of several days—annealing it—new peaks form. Following the

methods of Chapter 3, the locations of the new peaks can be used to show that the

copper and gold have formed a crystalline compound in which gold and copper are

arrayed as shown in Figure 5.4, called a

superlattice.

Many other combinations of

metals produce similar superlattices, usually when mixed in ratios of 1:1 or 3:1.

Figure 5.4. (A)

3:1

mixture of copper and gold has the equilibrium superlattice structure

depicted here below a temperature of

400

°C.

(B) Superlattice of copper and gold in equal

mixtures.The lattice constant c is 7% smaller than a.

The structures of superlattices are no less varied than the structures of inter-

metallic compounds, and all the crystal structures of Chapter 2 are possible. There

is no distinction in principle between intermetallic compounds and superlattices,

except that the latter lose their order at a definite transition temperature well below

the melting point of the crystal. Among elements mixed in 3:1 ratios, the structure

displayed in Figure 5.4(A) is common, while for elements mixed in proportion

of 1:1, the CsCl structure (Figure 2.8) frequently occurs, as in the cases of Cu-

Zn (/3-brass), CuBe, CuPd, AgMg, AgZn, AgCd, AuNi, NiAl, and FeCo. Phase

transformations in superlattices are studied in more detail in Section

24.5.1.

Many other structures are, however, possible. Mixed in a ratio of 1:1, copper

and gold form a superlattice involving alternating planes of copper and gold atoms,

as shown in Figure 5.4(B). When the atoms arrange themselves in this way, the

whole lattice contracts by 7% along the c axis, so the symmetry of the crystal

changes from cubic to tetragonal. Such macroscopic changes in the shape of the

crystal are another frequent feature of superlattices.

5.3.4 Phase Separation

Iron carbide, Fe3C, is a stable compound. Suppose that only 3 atomic percent of

carbon is mixed in with iron. The 3 carbon atoms out of 100 can combine with 9

Alloys

105

iron atoms to form 12 atoms' worth of Fe3C, leaving 88% iron. This process re-

quires the physical separation of

the

proper mixture of carbon and iron from the rest

of the soup, a rearrangement that cannot always occur quickly or spontaneously.

Because phase separation is based upon principles of thermodynamics much

more general than their application

alloy systems,

slightly abstract view

appropriate.

Suppose one has any two substances, whose free energy when they are mixed

homogeneously among one another has the form shown in Figure 5.5 as a function

of their relative concentration. The shape shown in the figure is all the information

one needs to conclude that

system prepared with concentration between c

and

c/,

will attempt to phase separate in order to minimize its free energy.

0 Concentration c

Figure 5.5. When the free energy of a homogeneous mixture of two substances has the

form shown in this diagram, whenever

the

overall concentration lies between c

and c/„ the

system will phase separate into components with those two concentrations.

Phase separation lowers the free energy of a system in the following way. Sup-

pose that the curve displayed in Figure 5.5 is iF(c). If the atoms were to separate

into two different domains of concentrations c

< c and

c (not yet necessarily

the concentrations indicated in the figure), the free energy would instead be

= f3

) + (l-f)?(c

), (5.22)

where

is the fraction of the sample at concentration c

, and

1 —

is the fraction at

concentration

Q,.

The fraction

is not arbitrary, because the overall concentration

of the mix must be c, and therefore

c =

(l-f)c

^f=^^-

(5.23)

Ca-Cb

?ps

^^Hc

)

^-2(c

(5.24)

The geometrical interpretation of Eq. (5.24) is that one picks any two points one

wishes on the curve 3(c) and draws

straight line connecting them; the straight

106

Chapter 5. Beyond Crystals

line describes free energy of phase separation between those two points. The points

and

in Figure 5.5 have been chosen so that this construction resulted in the

lowest possible free energy.

A typical phase diagram is largely occupied with regions of phase separation.

Consider the diagram in Figure

5.3(B)

for copper and silver. Regions Ag and Cu

are primary phases of fee metal with substitutional impurities. The region marked

"Liquid" is also a homogeneous mixture. Everywhere else, the two metals phase

separate. The solid lines then indicate the concentrations c

and

as a function of

temperature. For example, at 700

°C,

separation occurs between the two concen-

trations that have been marked. In the region denoted "Ag+liquid" a primary alloy

of silver coexists with a liquid containing a greater percentage of copper. The line

describing the composition of the liquid is called the liquidus, while the line de-

termining the composition of the solid is the solidus. The point marked "Eutectic"

has technological significance, since it provides the lowest possible temperature at

which the two metals mix in the molten state. As soon as the two metals are cooled

below the eutectic temperature, however, they begin to phase separate, so if the

goal is a homogeneous material there is a race against time.

Binary phase diagrams have a number of characteristic shapes that appear

bizarre at first glance, but which have a rather natural explanation in terms of sim-

ple assumptions concerning the free energies of solid and liquid phases. The idea

is best illustrated geometrically and is shown in Figure 5.6.

5.3.5 Nonequilibrium Structures in Alloys

A material composed of large numbers of small crystalline regions of different

orientations is said to be built out of

grains,

and the interfaces between them are

grain boundanes. These boundaries may appear on scales ranging from tens of

nanometers to meters. When the crystallites are at the small end of the scale, one

calls the material microcrystalline. Frequently in metals, the crystalline regions are

on the micron scale, and the materials are called polycrystalline. In sea ice, grains

may grow to scales of

meters.

The orientations of adjoining crystalline regions are

fairly random, and if one takes a two-dimensional slice through such a solid, the

grain boundaries form a characteristic network, with the grain boundaries meeting

in vertices, as shown in Figure 5.7. The manner in which grains grow is the subject

of Problem 6.

A type of grain boundary that occurs in substitutional alloys is the antiphase

boundary, which is a grain boundary where the orientations of the crystals on the

two sides are the same, but there is a shift of phase in the lattice as one crosses

the boundary. Antiphase boundaries can form snaking labyrinthine structures of

great complexity [Figure 5.8(A)]. As the concentration of one element in another

increases, one tends to get crystals dominated by one element embedded in crystals

dominated by the other element. For example, one can have small crystals of N13AI

sitting in a background of nearly pure Al. The forms that these intermixed crystals

can take depend upon the dynamical processes by which they form. In the simplest

Alloys 107

At sufficiently high tempera-

tures,

the liquid phase is of

lower free energy at all concen-

trations c than the solid.

At this temperature, the liquid

L is lower in energy to the left,

but coexists with solid of type

0 towards the right, and 0 is

stable for sufficiently high con-

centrations.

Now solid of type a is stable

for low values of c, 0 is stable

for high values, liquid is stable

for a small range in the middle,

and there are two coexistence

regions.

Only solid phases are stable.

These can be pure a, pure 0,

or mixtures a + 0 of the two.

Figure 5.6. Schematic free energies of liquid and solid which would lead to typical binary

phase diagram with a eutectic. The upper panels show solid and liquid free energies at

various temperatures, while the lowest panel shows the resulting phase diagram. In this

schematic view, the effect of temperature is simply a vertical shift in the relative free en-

ergies of solid and liquid phases. In reality, the shapes of the free energy curves would

change with temperature as well. Unless the solid at temperature T\ is metastable, there

will be no operational way to determine the solid free energy curve at that temperature.

108 Chapter 5. Beyond Crystals

Figure 5.7. Transmission electron micrograph of the grain structure of alumina. Grains

are approximately

μπι across. [Source: B. Hockey, National Institutes of Science and

Technology.]

case,

one has spheres of one type sitting in a background of the other type. The

spheres can be replaced by rods or plates, or can be sufficiently dense that they form

an interconnected network. The structures can include treelike dendritic shapes

[Figure 5.8(B)], arrays of parallel fingers, and bands or stripes.

5.3.6 Dynamics of Phase Separation

Any alloys heated sufficiently form a homogeneous liquid mixture. Upon cooling,

the mixture will typically remain homogeneous for a time, even if a phase sepa-

rated state is ultimately of lower free energy. In some cases, the initial process

of breaking apart into spatially separated regions of different phases happens eas-

ily, because the homogeneous state is unstable, and the result is called spinodal

decomposition. It can also happen that the cooled homogeneous state is stable

against small fluctuations, and only large rare fluctuations can disturb the situation.

The appearance of new domains by such fluctuations is called nucleation, and is

reviewed by Wu (1997) and Kelton (1991). As shown in Figure 5.8, phase sepa-

ration during the cooling of solid solutions can produce exceedingly complicated

spatial patterns. The patterns have an intrinsic aesthetic appeal, although from a

technological point of view they are usually undesirable, and one reason to under-

stand the physics underlying them is to prevent them from occurring.

The basic equation underlying the dynamics of phase separation is simply the

law of diffusion, presented in Section 5.2.2. The change in concentration with time

following from Eq. (5.1) is

β The concentration c is often taken to be a linear func-

Z__ _ 'Y)\7^r tion of actual atomic concentrations; for example, when Γ5 25Ì

ßf phase separating between c

and

Q,,

one might take c

—>

\ · /

(c-C

)/(c

-C

The diffusion equation, (5.25), appears innocent, but, when coupled with ap-

propriate boundary conditions, it is capable of producing the sort of complexity that

appears in Figure 5.8. As an illustration of how it functions, consider the problem

of a spherical droplet of

carbide,

carbon concentration c

, growing in a background

Allovs 109

Figure 5.8. (A) Antiphase boundaries in FÎ76AI23. The boundaries mark the dividing

line between two chemically identical regions, but where the placement of atoms is out

of synchrony. [Source: Allen and Cahn (1979), p. 1093.] (B) Dendrite formed during

solidification of stainless steel (F^oNiisCris). (Courtesy of L. A. Boatner, J. Gardner, and

Corrigan, Oak Ridge National Laboratory. )

mixture of iron and carbon whose overall carbon concentration is c^ > c

. Car-

bon atoms flow toward the droplet, because phase separation is thermodynamically

favorable. The simplest calculations employ the quasi-static approximation. This

approximation requires the time it takes for carbon atoms actually to join the drop

and make it grow to be long compared to the time for the concentration of carbon

outside the drop to converge to a steady state. In this case, the time variation of

c(r, t) can be neglected, and Eq. (5.25) becomes Laplace's equation

V c[r) = 0 Place the origin of a system of polar coordi- (5.26)

nates at the center of the droplet.

with boundary conditions

c(R) = c

and lim c(r) — c^. (5.27)

The unique solution of Eq. (5.26) satisfying (5.27) is

c{r) = c

+ -(c

-c

) (5.28)

and from Eq. (5.1) the flux of material into the drop is

ATTR

T>——

-. Take the gradient of Eq. (5.28) at« = r, and (5-29)

R multiply by the surface area of the drop.

Suppose that for each unit of concentration entering the drop, the volume of the

drop changes by v. Then the drop changes in size according to

=v4irR'D(c

-c

)

(5.30)

• VD,

=>R=—{coc-c

) (5.31)

RoiJ2vT>(c

-c

)t. (5.32)

110 Chapter 5. Beyond Crystals

The radius

the drop grows

as the

square root

time.

This calculation oversim-

plifies even

the

problem

growing drops.

neglects

the

fact that

the

surface

each drop exerts

pressure

on the

material inside, like

the

skin

of a

balloon, that

causes very small drops

shrink rather than grow.

neglects

the

fact that after

many drops have grown,

the

background concentration

CQO

carbon must dimin-

ish. Taking these effects into account, Lifshitz

and

Slyozov (1961) showed that

the

average size

drops increases

as the

cube root

time,

not the

square root.

The reason that

Eq.

(5.25)

able

exhibit complicated behavior

that

the

shape

of the

region where

the

equation applies

being made

depend upon

the

concentration field;

the

coupled motion

of the

boundaries

and

concentration field

complicated nonlinear problem. Further discussion

such pattern-forming

problems

provided

Langer (1980).

5.4 Simulations

the

power

computation increases,

it is

gradually becoming possible

imag-

ine computing

the

evolution

structures such

shown

Figure

5.8,

either from

partial differential equations

or by

starting down

at the

atomic level.

Two

impor-

tant methods

for

atomic scale calculation

are

Monte

Carlo

and

molecular dynam-

ics.

They

are

very similar

conception. Each method treats atoms

classical

interacting particles,

and

each watches them evolve

in a

fashion meant

mimic

the actual evolution they might display

nature.

The

methods differ only

in the

detailed manner that they employ

move atoms about.

5.4.1 Monte Carlo

The Monte Carlo method shuffles atoms about randomly, like

the

cards

in the

casi-

nos from which

takes

its

name.

The

idea behind

the

method

that

if a

solid

at some temperature

then

the

probability

its atoms adopting

position

that

their energy

is E is

exp[—/?£], where

ß = 1/kßT. As a

corollary,

follows that

one

has any two

states

the system whose energy differs

by SE,

then

the

relative

probability that these

two

states

occupied

exp[—ßSE}.

So suppose one has

collection of N atoms

. .

?N,

and an

energy function

E{7\

... ?/v).

What exactly this energy functional might

be is

discussed further

Section

11.8. In brief, the

Monte Carlo method chooses

atom randomly from

this collection,

and

randomly moves

it a

small distance.

If the

result

this small

move

is to

lower

the

energy

of the

system,

the

move

always accepted.

If the

move raises

the

energy

the system

by an

amount SE, then

the

move

accepted

with probability exp[—ßSE}.

Suppose

one

wants

compute

the

average value

thermal equilibrium

any

function

g(r\ . . . r#) of

particle positions.

The

function

could

be the

energy,

anything else.

The

Monte Carlo method proceeds

follows:

Simulations 111

Begin with a collection of particles whose locations are known and whose

energy E(r\ ... r#) = £1

nas

been computed. Compute g(r\ . . . r#) and

store the result in a variable G.

Choose one of these particles at random. Call the particle chosen particle /.

Create a random displacement vector. One way to do this is to choose three

random numbers p\ . . . pi lying in [0, 1] and to form the vector

Ξ =

2a{

- 1/2, p

- 1/2), (5.33)

with a setting the length scale. A natural choice of a is a typical interparticle

spacing, although final results should not depend upon the choice of a.

Compute the energy difference,

δε = ε(?

...η+Ε...Γ

)-ε

. (5.34)

When particles interact only with near neighbors, it will always be possible to

compute this change in energy more efficiently than by computing the energy

from scratch for all particles in the system.

Check whether δΖ is positive or negative. If

<5£

is negative, replace r/ by r/ + Ξ

and return to step 1. In either case, add the new value of

g(T\

. . .

?N)

to G.

6. However, if

δΕ

is positive decide randomly whether to allow particle i to move

or not. Pick a new random number p E [0,

1],

and compare p to exp[—βδ£]. If

p is greater than this Boltzmann factor, then leave ?/ where it is and return to

step 1. However, if p is less, set ?/ to η + Ξ before returning to step 1, despite

the fact that this move raises the energy of the system. Once again, add the

new value of g(r\ . . . r#) to G.

At the very end, after M steps of the process, an estimate of the thermal average of

g(r\ . . .

7N)

is given by taking the variable G that accumulated the sum of g and

computing

At low temperatures, almost the only moves accepted are those which lower the

energy of the system. At very high temperatures, almost every move is accepted.

The probability of accepting a move with positive δ£ has been arranged in just

such a way that the probability of occupying states differing by δΕ is exp[—βδ£].

This fact is easiest to understand by considering a system with just one particle,

and which can occupy just two states, but is true generally.

112

Chapter 5. Beyond Crystals

5.4.2 Molecular Dynamics

Molecular dynamics makes use of Newton's laws and random forces, rather than

random hops, in order to emulate thermal equilibrium. One form of molecular

dynamics operates by computing

Fi = -^r, (5-36)

the force on every particle due to every other particle, and then has the particles

move according to

fill

— f, mi is the mass of particle /. (5.37)

Practitioners usually want to use the largest number of particles that can possibly

fit on their computers, and they also want to obtain reasonable accuracy while

minimizing intermediate storage. For this reason, an algorithm due to Verlet (1967)

is widely employed; a variety of other algorithms is described by Rapaport (1995).

Pick a time step dt that is much smaller than any time scale on which forces would

cause particles to move appreciably. Let the position of particle / after n steps of

the algorithm be rf. In order to find positions at the n + l'st step, compute

=2rf-r?-' + ^/

(5.38)

with

Pp = P

{7{r

...r

) (5.39)

This sequence of computations requires only the storage needed to hold rf and

rf~ ', but makes errors only at order (dt)

after each time step, as shown in Problem

Nothing more than Newton's laws is required in order to carry out computer

simulations of a physical system at temperature T. However, it is not easy to know

what the temperature of the system will be before the simulation begins, because

while the total energy £ is easily specified in initial conditions, the temperature

would have to be found later—for example, from the root mean square velocity of

particles. The fluctuation dissipation theorem may be used to modify the method

so as to specify temperature directly. According to this theorem, interacting parti-

cles head toward equilibrium at temperature T when two terms are added to their

equation of motion, one a damping term, and the other a random force, giving the

Langevin equation

p The damping constant b is somewhat arbitrary. The

-jt _ i_ _ iy-γ

_i_?/f\

smaller it is, the more closely particles follow Hamil- /-c ΛΓ\\

' >

/ tonian mechanics, while the larger it is, the more ^ ' '

quickly they come into thermal equilibrium. De-

tails of a particular physical problem are needed to

... fixé.

with