Marder M.P. Condensed Matter Physics

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Problems

Figure 4.15.

cluster produced by diffusion limited aggregation with 2000 particles.

Diffusion-limited aggregation: Diffusion-limited

aggregation

or DLA is a

simple algorithm for how surfaces might grow in a disordered fashion, first

studied by Witten and Sander (1981). The algorithm has the following steps:

(a) Consider a rectangular grid, periodic in the horizontal direction, with period

200.

Particles will be able to visit nodes on the grid, indexed by integers

, ly), where 0 < l

< 200 and 0 < l

. Create a 200 x 600 array of integers

to represent this grid, and initialize all elements to 0.

(b) Introduce a particle at site (100, 0), which means setting the array element

(100,

0) to 1. The height of the cluster is now 1.

an array element on the line (l

, 1) and setting it to value 1.

(d) Allow this particle to diffuse. That is, choose randomly from the four direc-

tion vectors (1, 0),

(—1,

0), (0, 1) and (0, —1), and move the particle as the

direction vector indicates by setting the array element where the particle used

to be to 0, and also setting the value of the new array element to 1.

(e) Keep allowing the particle to diffuse until:

i. The particle height l

rises above the maximum height of the cluster (now

equal to 1). In this case set its array element to 0, and return to step (c).

ii.

The particle height hits the floor at l

—

0. In this case, the array element

where the particle is located remains at 1, and a new particle is introduced.

iii.

The particle occupies any lattice site adjacent to the first particle that was

introduced, from any side. Again, the second particle is frozen, and a new

particle will be introduced. There is now the possibility that the height of

the cluster will have risen to 2.

(f) Introduce more particles in an analogous fashion. Particles are introduced

randomly at height /

max

+ 1, where /

max

is the highest occupied site on the

existing cluster. If they diffuse above the height at which they are introduced,

94 Chapter 4. Surfaces and Interfaces

throw them out, and introduce a new particle. If a particle touches the floor,

or any particle already frozen in the cluster, it too freezes, /

max

is increased by

1 if necessary, and a new particle is introduced.

(g) Using this algorithm, produce a cluster with 2000 particles and draw a pic-

ture of it (see Figure 4.15).

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Today,

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24-31.

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Beyond Crystals

5.1 Introduction

The number of possible equilibrium crystal structures is enormous, but these struc-

tures represent ideal cases, rarely realized in the natural world. Macroscopic sam-

ples of solid matter are rarely close at all to thermal equilibrium, and equilibrium

structures are not always crystals. The goal of this chapter is to discuss some of the

relations between naturally occurring materials and crystals. The first topic is

dif-

fusion, since random displacement of atoms provides one of the primary ways that

perfect crystalline order is modified. Metal alloys provide the next topic. Simple

statistical arguments show that their crystalline order can be destroyed at a critical

temperature well below melting, and that even when one disregards this transition,

these alloys are unlikely ever to be in equilibrium. There follows a brief discussion

of several other forms condensed matter can take—liquids, glasses, liquid crystals,

and polymers—for which the language of crystals is inappropriate. The discussion

closes with an introduction to quasicrystals, which provide an example where a

structure without true crystalline order can be described with the same precision

and completeness that has been found for crystals.

5.2 Diffusion and Random Variables

5.2.1 Brownian Motion and the Diffusion Equation

In 1827, Robert Brown published an account of observations through a microscope,

beginning with pollen, and proceeding to a large variety of

powders.

He concluded

[t]hat extremely minute particles of solid matter, whether obtained from

organic or inorganic substances, when suspended in pure water or in

some other aqueous fluids, exhibit motions for which I am unable to

account and which from their irregularity and seeming independence re-

semble in a remarkable degree the less rapid motions of some of the

simplest animalcules of infusions. —Brown (1827), p. 481

The motions were eventually explained by Einstein (1905) and Perrin (1909)

as the result of thermal fluctuations causing fluid molecules to collide with the par-

ticles,

and exciting them into motion. The microscopic motions became identified

with the tendency of small particles to spread apart, as when some ink drops onto

a piece of paper or a gel.

Condensed Matter

Physics,

Second Edition

by Michael P. Marder

98 Chapter 5. Beyond Crystals

5.2.2 Diffusion

Diffusive motion concerns a large population of noninteracting particles kicked

about by random forces. Equivalently, it describes the probability distribution of a

single particle undergoing random motion.

The mathematical description of diffusion begins with the observation that if

n(r, t) describes some density distribution of particles then there is a particle cur-

rent j that flows from regions of high density to low. The simplest possible rule

for how this happens is that the rate of flow is proportional to the gradient Vn(r, t)

(Fick's law) so that

j = -VWn. (5.1)

The constant Ί) is called the diffusion constant. Eq. (5.1) can also be rewritten as

j = —vn where v =—T>— (5.2)

is the mean velocity of particles.

Continuity Equation. To obtain a closed expression for the particle density n,

make use of the fact that particles are conserved. The continuity equation, illus-

trated in Figure 5.1 describes any collection of conserved particles whose coordi-

nates r change continuously with velocity v. If n(x, t) is the number of particles

at position x and they are moving with mean velocity v(x, t), then the number of

particles moving into a little region of width dx minus the number moving out is

Av(x)n(x) dt—Av{x + dx)n{x + dx) dt (5.3)

in time dt. Therefore for variable x one has that

dn d , . . .

= V(x)n(X t). Divide change in number of particles per time (5.4)

dt dx ' by volume Adx of the box.

If there are many variables, the argument holds for each coordinate in turn and

dn ^ d

~dt

~^dr,

= - Σ ^-Mr)n(7, t). (5.5)

Figure 5.1. The continuity equation ap-

plies to any situation in which a conserved

collection of particles can

characterized

by a mean velocity wasa function of co-

ordinate

Diffusion and Random Variables 99

Diffusion Equation. Using the velocity v for diffusing particles from Eq. (5.2) in

the continuity equation (5.5) gives finally the diffusion equation

— DV n. Taking Eq. (5.5) to apply to three dimensions. (5.6)

5.2.3 Derivation from Master Equation

The simplest derivations of the diffusion equation do not make it apparent how

random motion of particles leads to diffusion. To build this connection, a more

sophisticated derivation is required. Again, let n(7, t) be the density of particles

at location 7 at time t. Assume that at small time dt later the number of particles

originally in small volume 5V at

that end up at 7 + ?

n(r, t) R(7

—>

7 + i, dt) ÔV.

ives tne

probability that particles start at 7 (5.7)

and end at r + ? after time t.

Then the change in the number of particles in the volume 5V near 7 is the number

of particles starting at

and coming to 7 minus the number starting at 7 and

leaving, which can be written as

n(7, t + dt)— n(7, t) = / ds -

R(7

s-^ 7, dt) n(7+s, t)

■

R(7

—>

r-s, 8t) n(7, t)

(5.8)

Factors of 8V have been divided out everywhere. Equation 5.8 is called the master

equation. It appears very general, but it is based on some strong assumptions that

can easily be violated. The most important is that knowing a particle is at 7 is all

one needs in order to know the probability it will be at

+ s some time dt later.

This need not be true. For example, if the momentum of particles is important,

then a collection of particles with momentum p\ will have a different probability

distribution for future locations from particles with momentum ριφ p\- On the

other hand, nothing in the derivation requires ? just to be three position coordinates.

The derivation goes through without change when 7 contains many components,

including for example momenta as well as positions.

To derive the diffusion equation, observe that /?(?+?—>

t should be expected

to be a rapidly varying function of

the chance of going distance s varies quickly

with the distance. However, one should not expect R(7 +

—>■

7, t to vary very

quickly as a function of r: the rate at which particles jump about should not depend

very much on where they are. Adopting this assumption, one can insert a Taylor

expansion in Eq. (5.8) sending

7 —

R(7+67^7, 6t)n(7+s, t) « R(7->7-s, öt)n(7, t)

07-

—R(7—+ 7 —

Ï, ôt)n(7, t)

+ \ Σ

ßx R{7^7-s, dt)n(7,t) +

. .

αβ

ΟΓαΊΓ

(5.9)

100

Chapter 5. Beyond Crystals

Substituting Eq. (5.9) into Eq. (5.8) gives

-^ =

- Σ

ΈΓ

«('> 0 +

äryVoß

^')>

10)

<Jr

where, sending ?

—►

— sin the integrals,

V) =

fdss

ldt)

(5.11a)

aß

(r)

J ds\

dt)

(5.11b)

The diffusion equation has now been generalized to include the possibility of a

nonzero net flow «ina particular direction, and the possibility that the diffusion

rate is different in different directions so that Ί) becomes a tensor.

5.2.4 Connection Between Diffusion and Random Walks

Using Eqs. (5.11), one can connect physical models for how particles move to the

diffusion equation. For example, suppose that at time intervals io, particles jump a

distance a in a random direction in three dimensions. The mean flow w vanishes,

because the probabilities of going forward or backward in direction a are equal.

Off-diagonal components of Ί) vanish as well. Since jumping distance a in time

is certain,

R(r^r + s,to) =

-^^-. (5.12)

The normalization ensures that

dsR= 1. Thus

Σ©

Ωα

(7)=/

[

2^(

)

—

(5.13)

Ans

to 2/o

=>·

D =

Έαα

Since the three components are equal by symmetry (5.14)

6io

Thus a random walk with step length a and time constant to is equivalent to a

solution of the diffusion equation with diffusion constant D = α

/6ίο· This cor-

respondence can be used to find the mean square distance travelled by particles

undergoing a random walk. Multiply the diffusion equation (5.6) by r

and inte-

grate:

drr ~—-=T)

/ dr r ^rn =

6T> Use spherical coordinates for

and (5.15)

V Qr assume n(r, t) is spherically symmetric.

Then integrate twice by parts and use the

2\ ^m> 2^

fact that

drnCr, t)

1. ...

. -.

) =6T>t =

■>

(5.16)

Alloys

101

5.3 Alloys

The importance of metallic alloys is compactly expressed in the fact that eras of

human history are named for them. The Bronze Age began around 3000 B.C. in

Greece with the discovery that addition of between 10% and 30% of tin to copper

produced

metal of greater hardness and lower melting point than either of its

components. Bronze was in most respects superior to the alloys of the Iron Age

which followed, except that iron was cheaper and easier to mold. The progress of

industry in recent centuries has closely been tied to the creation of improved forms

steel,

which meant developing ingenious ways to add carbon and other materials

in a controlled fashion to iron.

5.3.1 Equilibrium Structures

Consider

pure crystal of some element—say iron—and imagine addition of a

very small amount of

second element—say carbon. The second element is always

soluble in the first in sufficiently small quantities.

The reason that thermodynamics favors a mixture is a consequence of entropy.

Suppose that addition of each atom of the second element to the first incurs an

energy penalty e > 0. The number of ways to add M atoms to a lattice of N 3> M

sites is

/N\

N\ N

Using the approximation that

(5 17)

\MJ M\{N-M)\ M\ ' N(N-l)... (N-M+\)^N

so the entropy associated with the addition of M atoms, in terms of the concentra-

tion

c = M/N, (5.18)

kß in(N

/Ml) « —ksN(c inc

—

c).

The

entropy is Boltzmann's constant kg (5.19)

times the log of the number of ways to

achieve some macroscopic state; also use

M\?sM

Therefore, the free energy

of the mixture is

= £

TS =

N[ce

+ k

Tc In c

Tc], (5.20)

which has a minimal value for concentration

C~e

\ Just differentiate Eq. (5.20) by c, set to zero, (5.21)

and solve for c.

This tendency toward solubility of one element in another that declines exponen-

tially with decreasing temperature is illustrated in Figures 5.2 and 5.3(B).

102 Chapter 5. Beyond Crystals

IL,

-4 -

^ -5 -

Figure 5.2. The solubility of

Fe3C

in Fe drops exponentially, as predicted by Eq. (5.21).

[Source: Flynn (1972), p. 38.]

Zone Refining. An immediate consequence of Eq. (5.21) and diagrams such

as Figure 5.3 is that the solid elements will not naturally be found in completely

pure form, but tend to contain impurities at the level of around 1%. This tendency

is particularly unfortunate in the case of the semiconductors intended for use as

electronic materials, because certain types of electrically active impurities impair

device operation if they are present at levels over one part in 10

. The same

physical principle that tends to introduce impurities into solids can also however be

used to remove them, motivated by the observation that as temperature increases,

the solubility of an impurity in a host increases. Zone refìning uses this idea by

taking an impure crystal and heating it, starting at one end, and slowly moving

the front of the heated region toward the other end. Because impurities are much

more soluble in the hot material than in the cold, they migrate toward the hotter

end. When the process is completed, one end of the crystal has a much higher

concentration of impurities than the other. The portion highest in impurities is

discarded, and then the process is repeated. With successive passes, the impurities

can be filtered away.

5.3.2 Phase Diagrams

The material produced by addition of a slight amount of one substance into another

is called a primary alloy, since the first material is essentially unchanged by addi-

tion of the second. The primary alloys come in two basic types. In a substitutional

alloy, one atom displaces the other and lives on its lattice sites. Zinc mixed into

copper provides an example of an alloy of this type. Alternatively, the additional

atoms may choose to sit within empty spaces of the lattice, creating an interstitial

alloy; the atom sitting within interstices must be small and is usually hydrogen,

boron, carbon, or nitrogen.

A phase

diagram

describes the equilibrium state of a mixture of elements, as a

function of temperature and relative concentrations. The simplest two-component

diagrams belong to a few pairs of elements, including AuAg, AuPd, NiMn, CuPt,