Marder M.P. Condensed Matter Physics

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Two-Dimensional Dislocations and Hexatic Phases 393

Experimental Observations. The two-dimensional crystals in which bond-ori-

entational order is observed would be triangular lattices at zero temperature, and

therefore they are known as hexatics. It is not easy to observe hexatic ordering

in atomic scale systems, because free-standing films one atom thick are hard to

make. It has been observed in liquid crystal films only a few monolayers high by

Pindak et al. (1981). Bond-orientational order has also been observed in colloidal

suspensions, where latex spheres of about a micron in diameter are placed in a

micron-thick bath of water. The spheres interact through electrostatic repulsion.

Figure 14.12 shows experimental scattering data obtained from such a colloidal

system. The left panel shows the colloidal system heated above its melting tem-

perature and displaying the correlations typical of a liquid. The center panel shows

the system in an intermediate phase, the hexatic liquid in which orientational order

is still present, but no traces of positional order remain. The right panel shows the

hexatic crystal, which is the phase described by calculations in the preceding sec-

tions.

The lack of long-range positional order is indicated by the nonzero widths

of the scattering peaks, along with their slow decrease in amplitude away from the

upper right-hand corner. Hexatic orientational order, however, is still perfect, so all

the peaks are in the location one expects for a perfect crystal. The two-dimensional

positional ordering of this structure is called quasi-long-range order.

Liquid Hexatic Crystal

t = 0s

r = 0.05s

i = 0.1 s

Figure 14.12. Experimental correlation functions g(r, t) from a two-dimensional colloidal

system. The top panel shows correlation functions at t = 0, while the lower panel shows

the decay of correlations after 0.05 and 0.1 s. [Source: Murray and Grier

(

1996).]

394 Chapter 14. Dislocations and Cracks

Figure 14.13. A two-dimensional array of parallel screw dislocations.

In the colloidal crystals, particles are confined to a plane, and they move in the

x and y directions. The dislocations that arise are edge dislocations, as in Figure

14.5.

To study their statistical mechanics, two simplifications will be introduced.

The problem will be treated in the framework of linear elasticity. Continuum

mechanics only runs into trouble near the core of a dislocation, and it will be

sufficient to cut off certain dangerous integrals, as in Eq. (14.47).

The vector character of edge dislocations makes them hard to work with, so

they will be replaced with screw dislocations, as in Figure

14.13.

Thus, in-

stead of permitting points in the plane to have displacements u

, u

, for the

remainder of the section they will be allowed only displacements u

, which

will be denoted by u. This alteration of the problem simplifies the analysis,

but does not change any of the qualitative conclusions.

The displacement fields being considered from now on are of the form

= 0,

= 0, u(x,y) = u

(x,y). (14.48)

According to Eq. (12.33), the energy of

displacement field obeying Eq. (14.48)

U = -tl I d r (S7u) The gradient operator is understood to be op- (14.49)

2 J erating in two dimensions. The factor a out

front is needed so that the dimensions of U

will turn out correct; think of

the

solid of hav-

ing height a in the z direction.

According to Eq. (12.40), such a displacement field in static equilibrium obeys

M = 0. Laplace's equation! (14.50)

Single Dislocation. The simplicity of

Eq.

( 14.50) makes it possible to write down

immediately the displacement field of a single dislocation. All analytic functions

of a complex variable are solutions of Laplace's equation, so it is sufficient to find

within the repertoire of such functions one that has the property of increasing by a

Burgers vector when one passes in a complete circuit around the dislocation. The

Two-Dimensional Dislocations and Hexatic Phases 395

logarithm is the simplest complex function this property; \og(x + iy) increases by

2πί for any path that loops once about the origin. A dislocation at the origin with

Burgers vector a is described by

u(x,y)

=—lmln[x

iy\.

(14.51)

Ζ7Γ

What is the potential energy cost of introducing such a dislocation into a solid?

Placing Eq. (14.51) into Eq. (14.49) gives

-τ(έ)7<Η-^Γ+[^]

an ( a \

I I / dr

ΊτΤΓ—

The limits a and R are introduced by hand to (14.53)

2 \ 27Γ / Ja i keep

tne

integral from diverging, as discussed

below.

(aß) In I — 1 -\-W.

w K

added by hand to account for the energy (14.54)

47Γ y

) °f the core of the dislocation.

The passage from Eq. (14.52) to Eq. (14.54) is not completely direct. There is

the problem that the integral over r diverges both at large and short lengths. The

solution at large lengths is to cut the integral off at a system size R (taking the

dislocation to sit at the center of a cylinder). At short lengths, the integral must

be cut off at the scale of the lattice spacing a, where continuum mechanics breaks

down. To compensate for the very crude treatment of short lengths, it is necessary

to add to Eq. (14.55) an additional energy w, which describes the energy needed

to form the core of the dislocation, meaning those fine adjustments on the atomic

scale that

Eq.

(14.54) has gotten wrong. This energy w will be added to the energies

of all dislocations from now on.

Two Dislocations. The displacement field of two dislocations with equal and

opposite Burgers vectors separated by distance

along the x axis is

u(x, y) = — Im {ln[x +

/_y]

- In[X

—

+'iy}}

■

(14.55)

27Γ

Using the divergence theorem and some properties of complex functions, Problem

2 shows that the energy of such a configuration is

In (^) +2«, with q

= ^. (14.56)

\a J 4π

The notation of Eq. (14.56) is meant to suggest an analogy with two-dimensional

electrostatics. The analogy is almost perfect. The dislocations obey exactly the

same equations as lines of charge

The displacement u serve as the electrostatic

potential, and the gradients of u are equivalent to the electrical field. The advan-

tage of employing this analogy is that it permits the use of a subtle quantity, usually

taken for granted, the dielectric constant e. There is a slight flaw in the analogy,

which emerges as soon as one draws the field lines Vw that should correspond to

396

Chapter 14. Dislocations and Cracks

the electric field. As shown in Figure 14.14, the field lines surround the disloca-

tion in an axial fashion. This is the magnetic field configuration surrounding a line

of current, not the electrical field around an electric charge. However, because the

equations describing the energies of interacting line currents are identical to the en-

ergies of interacting charges in two dimensions, the analogy with two-dimensional

electrostatics survives and will be employed frequently in what follows.

# *> » * %

^ ..

—

«. \

• * / \ « ·

\ ». -» x /

% m ~ + *

Figure 14.14. Vector

field

for a dislocation described by Eq. (14.55). The

field

is the

same as the magnetic

field

around a line of current.

Statistical Mechanics of Two-Dimensional Dislocations. According to

Eq. (14.54), the energy of an isolated dislocation diverges logarithmically with

system size. Despite this divergence, thermodynamic equilibrium may happily

permit many such dislocations to form. All depends upon the competition between

energy and entropy. The entropy of forming a single dislocation may be estimated

by noting that in a square with side length L, there are

locations at which to

put the center of the dislocation, so its entropy is

S = 2k

ln(L/a), (14.57)

while its energy will be

Picking out the divergent part of the energy,

c _ „2 \

(T IQ\ and assuming that the difference between a ΙΛΑ <Q\

y v / / · disk of radius L and a square of side length L \ ■ )

is not important.

There should be a critical temperature at which it becomes favorable to flood the

system with dislocations. Using the estimates in Eqs. (14.57) and (14.58), the free

energy

—

TS becomes negative at a temperature

= ^. (14.59)

Two-Dimensional Dislocations and Hexatic Phases 397

This estimate of the transition is close to the truth, but the numerical value of the

transition point will turn out to be slightly different.

Kosterlitz-Thouless Hamiltonian. Motivated by Eq. (14.56), consider a popula-

tion of screw dislocations interacting with energy

K=-Y^U{\r

-r

\)+2w with U{r) = 2q

ln(r/a) (14.60)

At low temperatures, the dislocations group themselves into pairs that form a

dilute gas. Proof of this statement comes from comparing the mean square radius

of a dislocation pair to the mean square separation between pairs.

At low temperatures, the mean square separation of two dislocations in a pair

= ^

(14.61)

/•oc

/ dr

2-nr

poo

~ßq

ßq

-ßU(r)

2nre

-f

-2

-ßU(r)

Insert Ec

Insert Eq. (14.60) and perform the integrals. (14.62)

On average, the pairs are situated farther from one another than

). Con-

sider, for example, the density n of dislocation pairs whose separation lies between

r and

+ dr. The grand partition function for vortex pairs is

V-

(

,-ßU(\7

-r

\)-2ßw

ër

=\+^e-

l1u

^-^

. . .

(14.63)

The density n(r) of dislocation pairs with separation lying between r and r + dr in

a system of size R

is therefore, to leading order in exp[—2ßw],

<r)dr=^

)

^ E

e-^'-™-

^,,,,-*,

+ ·

· -(14.64)

% -r-

-ß

(

)-~

\ Total sites in system is roughly/?

· (14.65)

If the core energy w is large enough, n{r) can be made as small as desired for any

given r. At low temperatures where large pairs are unlikely, the typical distance

between dislocation pairs is therefore much larger than the separation between dis-

locations in the pair. However, as the temperature rises, the density of pairs rises,

and their interaction becomes important.

The essential idea of Kosterlitz, Thouless, and Berezinskii is to treat the in-

teraction of dislocation pairs through a dielectric constant. After all, what one has

here is a collection of dipoles of different sizes, which in thermal equilibrium rotate

through all angles so there is no spontaneous polarization, but which would orient

398

Chapter 14. Dislocations and Cracks

and create a polarization P in the presence of any electric field

This situation is

just what the theory of dielectrics was created to handle. In a dielectric medium, the

interaction energy of

two

charges is not U = 2q

\n(r/a), it is

ln(r/a)/e,

and so interactions between many different dipoles can be accommodated by using

instead of U.

However, there

an interesting complication. As the phase transition point

is approached, dipoles of larger and larger sizes are created, while simultaneously

the number of very small dipoles grows rapidly. The very smallest dipoles do not

know they are sitting in a dielectric medium because their charges are much closer

together than the distance to any other dipole. The appropriate dielectric constant

for them must be 1. Very large dipoles may have large numbers

small dipoles

in between them, and therefore they can have

dielectric constant e

1. So the

dielectric constant must be taken

depend upon scale. When the dislocations

become numerous, they can be assigned to dipoles by grouping every dislocation

with the nearest other dislocation that does not already have a partner, starting with

the closest dislocations and moving away

distance. The idea

constructing

a scaling theory in which the results

one length scale are formulated in terms

of the results

at a

smaller length scale

the basic idea

the renormalization

group. Similar ideas appear also in the study of metal-insulator transitions (Section

18.5.2),

of critical phenomena (Section 24.6), and in the study of the Kondo effect

(Section 26.6).

Adopting now completely the language of electrostatics (equivalently, magne-

tostatics), focus on

dislocation dipole of separation

but arbitrary orientation

Θ.

When placed in an electric field,

will tend to orient itself along the field.

The

polarizability a(r) of a dipole

of size

is given by

p —

aE =

rq((cOS

Θ, sin

Θ))

Taking

to point along 0

(14.66)

f d±

-ßU(r)-2ß

W+ß

rcos θ^^ ^

sm Q) {Η6η)

2π

—

ßq r E.

Expand the exponential in powers of

up to

(14.68)

linear order and perform average over

Θ.

So the contribution to the dielectric susceptibility

d\{r)

from dislocation pairs of

size between

and r + dr is, from Eqs. (14.65) and (14.68),

(r)

= n(r) dr a(r)

\ßq

(-Y

-ßvl<r)-2

ßw

69)

\aj a

In order to determine e, make the assumption that all dipoles smaller than

contribute to the dielectric constant screening the interactions

scale

and all

larger ones are irrelevant. This point is the crucial junction where results

one

length scale determine an effective theory

larger scales. So, remembering the

result from electrostatics that e

4πχ, one now has

e(r) =

l+4n

ί/χ=1+4π

dr'n(r')a{r')

(14.70)

Cracks

399

de(r)

de(x)

■

ßq

-ßU(r)/e(r)-2ßw

(14.71)

4π W-*·

3-2ßc,

/e(x)~2ßw

. Withx = r/a, and using Eq. (14.60). (14.72)

Solutions of Eq. (14.72) appear in Figure 14.15. At low temperatures, the di-

electric function e(x) goes to a constant for large x. At a critical temperature, the

nature of the solution changes dramatically, and e diverges on large scales. For

the electrostatic analog, the transition corresponds to a transition from insulator

to metal, because the system becomes infinitely polarizable, while for the original

system of dislocations, it corresponds to a melting transition in which the disloca-

tion pairs become unbound. The result is the destruction of quasi-long-range order

depicted in Figure 14.12.

-i

-3

/V =

^ßq

= 2A6

f ßq

= 2.20

W 7?<?

= 2.40

x = In r/a

Figure 14.15. Solutions of Eq. (14.70), for w = 2q

and for ßq

ranging across the tran-

sition point from 2.0 to 2.4. For temperatures below the transition, the dielectric constant

flows to a constant value at large distance scales, while above it the dielectric constant

diverges, signaling the destruction of quasi-long-range order.

14.4 Cracks

14.4.1 Fracture of a Strip

Ductile materials flow when placed under sufficient stress, but brittle materials

break, and the sort of defect that allows them to do so is quite different from the

dislocation. Brittle solids fail through propagation of cracks, which are macroscop-

ically long regions of separation, with a microscopically sharp tip. The question

400

Chapter 14. Dislocations and Cracks

Figure 14.16. The upper surface of

strip of height L and thickness w is

rigidly

displaced

upwards by distance δ. A crack is cut through the center of the strip, and it relieves all

stresses in its wake. When the crack moves distance dl from (A) to (B), the net effect is to

transfer length dl of strained material into a length dl of unstrained material.

of whether a given solid should be brittle or ductile is not easy to determine the-

oretically, and it is bound up with the question of whether a sharp crack tip will

propagate, or whether it will emit dislocations and become blunt, as reviewed by

Carlsson and Thomson (1998). Roughly speaking, when atoms are connected by

large numbers of bonds, individual bonds can easily be broken and reformed, dis-

locations move easily, so fee and bec metals such as copper, lead, or iron tend to

be ductile. The hexagonal metals, such as beryllium, are more likely to be brittle,

and covalent solids such as silicon or carbon tend to be very brittle.

An important feature of cracks is their ability to funnel energy from large ex-

panses of a macroscopic object, and to direct it down to the atomic scale in the

service of snapping bonds. The way this process works is quantified by a simple

scaling argument.

Consider a long strip with a crack down the middle. The upper and lower

boundaries of the strip are rigidly displaced upwards by distance δ, as shown in

Figure 14.17. For brittle solids under tension, energy

ΤΓ. ', ~1Γ and force as a function of displacement take the uni-

Displacement

ò _

_ , · , ,·

versai forms shown in these diagrams.

Displacement δ

Cracks 401

Figure 14.16. Far to the right of the crack, the potential energy U stored per unit

length is

U = —δ W—

is Young's modulus and

is the thickness (14.73)

2 L of the strip.

while far to the left there is no stored elastic energy. In Figure 14.16(B) the crack

has moved a distance dl. The elastic fields around the crack tip may be compli-

cated, but they are the same in (A) and (B), and the total change dU in the stored

elastic energy between the two cases is

dU = dl-5

w-. (14.74)

2 L

The only difference between (A) and (B) is that the crack has moved distance

dl. Suppose that it costs fracture energy Γ per unit area to extend the crack, and

suppose that all the elastic energy relieved by crack motion goes into creating new

crack surface. Then

i _ v

(14.75)

Tu> dl = dl-δ w —

2 L

δ=

and

Gyy

2ΓΥ

L '

(14.76)

Therefore the stress needed to make a crack propagate in this geometry is pro-

portional to the square root of Young's modulus and inversely proportional to the

height of the strip, meaning that large objects break more easily than small ones.

Consider now a strip without a crack in it at all. Place it under tension. Its

energy will rise according to Eq. (14.73) until enough energy is available to allow

a crack to run. Once the crack has severed the object, additional tension no longer

changes the energy. Note from Eq. (14.76) that the tension at which the crack can

initiate becomes smaller and smaller as the height of the system L becomes larger

and larger. Therefore, for large enough systems, the energy and force-displacement

curves for all solids take a universal form, as shown in Figure 14.17. In princi-

ple,

these curves represent the thermodynamic minimum energy of solids under

tension, with separation occurring when the energy per length equals twice the

surface energy 7. In practice, solids only act in this fashion when cracks are ini-

tially present and are able to run through them. A brittle solid is by definition one

for which the energy-displacement or force-displacement curves resemble Figure

14.17.

Equivalently, a brittle solid is one where cracks are able to run consuming a

constant amount of energy per unit area.

To understand why cracks are so effective in breaking brittle objects, it is help-

ful to realize that they concentrate stresses. If one takes a plate, puts an elliptical

hole in it, and pulls, then the stresses at the narrow ends of the hole are much larger

than those exerted off at infinity, as shown in Figure 14.18.

The ratio of maximum stress at the tip of the hole to the stress applied far away

Maximum stress / / ,

„

Oc \ — The coefficient of proportionality is of order (14.77)

applied stress y R

unity.

402

Chapter 14. Dislocations and Cracks

Figure 14.18. The stresses at the tips of

elliptical hole in a solid are much greater than

those applied off at infinity. The plot shows a solution of a

derived from

Eq.

(14.99) for

Σ =

and m = 0.9. Note the 20-fold increase in stress near the crack tip.

where / is the length of

the

hole and R is the radius of curvature at the

tip.

Therefore,

assuming that typical solids have cracks with tips of radius 10 Â and of length 10

Â, one can account for the discrepancies in Table 14.2. If thin wires of glass or

iron are carefully prepared so as to be absolutely free of surface flaws, then their

strength under tension in fact reaches the ideal limits listed in the table.

14.4.2 Stresses Around an Elliptical Hole

It is possible to find the stresses surrounding an elliptical hole sitting within a plate

that has been placed under remote tension, as illustrated in Figure 14.18, and to ver-

ify Eq. (14.77). To simplify matters as much as possible, assume that the stresses

applied to the plate are such as only to create a displacement u = u

in the z direc-

tion, which as in Eq. (14.50) obeys Laplace's equation,

w = 0. (14.78)

The theory of complex variables can once again be brought to bear in order to find

solutions. For the boundary problem at hand, conformai mapping

the appropriate

technique. Because u is a solution of Laplace's equation, it can be represented by

Φ(0

(14.79)