Marder M.P. Condensed Matter Physics

Подождите немного. Документ загружается.

Liquids 113

The brackets refer either to a time average, or to

(5.41 )

average over statistical realizations of the system.

See Landau and Lifshitz

(

1980) p. 362 or Problem

A practical implementation of this idea replaces F" in Eq. (5.38) with G" where

tyn_jf-l] i

Ö" = Ff-bmi^—

—

- + Êi^6bmik

T/dt; (5.42)

Ξ/ is a vector whose components lie randomly between —1 and 1, and it can be

computed from

Ξ = 2(ρι — 1/2 ΏΊ — 1/2. P3 — 1/2) Each p

is a random number between 0 and 1. (5.43)

Because molecular dynamics keeps track of both positions and momenta, it is

computationally more costly than Monte Carlo, and Monte Carlo is preferable if

one only wants to find thermodynamic averages for a system in equilibrium, be-

cause it runs faster. However, molecular dynamics paints a more realistic picture of

the dynamical fashion in which a system approaches equilibrium. Representative

results from molecular dynamics calculations appear in Figures 5.9 and 5.14.

5.5 Liquids

5.5.1 Order Parameters and Long- and Short-Range Order

Every element melts at some temperature, at which point crystalline order disap-

pears.

The presence and absence of order is captured by an order

parameter,

which

is a function designed to (a) vanish when the desired form of order is absent, and (b)

rise up from zero as soon as it is present. Order parameters are often single num-

bers,

although they may also be tensors. The sharp Bragg peaks that characterize

scattering from crystalline lattices can be used as crystalline order parameters.

Formally, to create an order parameter 0^ distinguishing between crystal and

liquid, choose any reciprocal lattice vector K ψ 0 of the crystal, look back to the

correlation functions defined in Section 3.5 and define

0 - = —n

(q = K,0). Where n

{q, t) was defined in Eq. (3.54). (5.44)

For a crystal where N

terms contribute to the sum in Eq. (3.54), this quantity

should be of order unity. In a liquid where crystalline order has been lost, it will

instead be of order

/N.

Radial Correlation Functions.

Comparisons of solids and liquids are often made by additionally averaging the

correlation function

«2(^1,

^2;

0) defined in Eq. (3.49) over all angular orientations

(uo)m)=™

BTSaßm

114

Chapter 5. Beyond Crystals

of the sample and making the result dimensionless by dividing through by the

square of the density n = N/V. The result is the

radial correlation

function

g{r)

(r\,r

;0))e δ(7

-7

)

Here r = \7\

—

τ\\. The angular braces mean

one must average over all angular orientations

of the sample. Subtracting off

the

second term

is conventional.

(5.45)

Liquids and glasses are isotropie by nature, while polycrystalline and powdered

crystalline samples have been made isotropie as described in Section 3.3.3. For

scattering off such materials the structure factor in Eq. (3.52) can be rewritten as

S(q) = \+n

drg[

iq-r

,+„/

dr(g(r)-\)é^

+ n

Using Eq. (5.45) and the static (5.46)

structure factor of

Eqs.

(3.55) and

(3.54).

dr e'Q'

Since g(r) ~^

for large r the (5.47)

integral only converges well after

subtracting 1.

\+n / dre^

{g{r)-\).

The last term on the right hand

side of (5.47) is a delta function

that only rises above zero when

one is staring directly into the

scattering beam, and which one

therefore can drop.

(5.48)

An integral of the area under the first peak,

first peak

dr 4πΓ g(r), (5.49)

gives the average number of nearest neighbors of each atom, known as the coor-

dination number. This quantity is slightly ambiguous to the extent that the precise

ending point of the first peak is ambiguous.

Figure 5.9 provides typical examples of correlation functions for crystals and

liquids taken from experiment and from computer simulation. The defining prop-

erty of crystals is long-range order (Section 3.5.1), yet the long-range order found

for crystals in these examples is not so very long: only around 10 Â. Nevertheless

order over these distances is easily sufficient to distinguish the liquids and crystals

from each other.

Figure 5.10 displays the static structure factors S(q) for liquid and amorphous

nickel. The radial correlation function g(r) can be obtained by inverting the Fourier

transform in Eq. (5.48). Differences between static structure functions of liquids

and amorphous solids are quite subtle.

5.5.2 Packing Spheres

One of the oldest questions in condensed matter physics goes back to a conjecture

of Kepler that the most efficient way to fill space with spheres is to stack them in

an fee or hep lattice (Figure 2.2 (D)). If one examines any given sphere in the close

packed state it has 12 neighbors, but these neighbors are not distributed about it

Liquids

115

Figure 5.9. (A) Radial correlation function g(r) for liquid and crystalline sulphur obtained

from neutron scattering. For nearest neighbors, the correlations in liquid and crystalline

sulphur are very similar. [Source: Winter et al. (1990) p. SA218.]. (B) Radial oxygen-

oxygen correlation function g(r) for liquid and crystalline water obtained from molecular

dynamics simulations. The crystalline water is in the form of Ice VIII at 10 K and pressure

of 2.4 GPa. [Source: Vega et al. (2005), pp. 1453 and 1455].

5ί

Figure 5.10. Static structure factors S(q) for liquid and amorphous nickel. The difference

between the liquid and amorphous states appears in subtle changes in the shape of the

second peak. [Source: Waseda (1980), p. 91.]

perfectly symmetrically. Problem 6 in Chapter 2 shows that such collection of hard

spheres fills 74% of space. Kepler's conjecture was proved after a slight delay of

several centuries by Hales (1997, 2005).

Spheres can also be mixed together randomly, in which case they provide a

model of a solid called dense random packing or the Bernal model. Bernal ( 1959)

carried out experiments with ball bearings and showed that randomly mixed hard

spheres fill about 64% of space. They do not instantly and automatically find their

way into the closely packed fee or hep structures. Furthermore, there seems to

be no way to fill space with hard spheres in a uniform way at densities between

116 Chapter 5. Beyond Crystals

74%

and 64%. Ensembles of hard spheres provide attractive settings for carrying

out computer simulations. The pair distribution function for hard spheres in two

dimensions is shown in Figure

5.11.

While there is no element that precisely repro-

duces the three-dimensional hard-sphere distribution function, it is not too far from

that for liquid argon. In addition, properties of randomly packed spheres provide

a starting point for studies of granular materials like sand [Liu and Nagel (2001);

de Gennes (1999); Kadanoff (1999); Jaeger et al. (1996)].

"öo

0 12 3 4 5

r/d

Figure 5.11. The radial distribution function g(r) for hard spheres (disks) of radius d in

two dimensions. The energy of

system of

hard

spheres is defined

be zero if

the

spheres

are not overlapping, and infinite if any two touch. Temperature is therefore irrelevant for

this system, and its correlation function depends only upon density, which in this case is

chosen to be 0.5.

5.6 Glasses

Just as ball bearings form a static random state when mixed together, so do many

collections of atoms. Solids where atomic positions are largely random are called

amorphous materials or glasses. Glasses are distinct from liquids. In liquids, the

atoms are constantly moving about and exchanging places, while in glasses they

are mainly locked into place. There is not complete agreement on how precisely

to define what a glass is, nor which elements are capable of forming glasses. No

elements or mixtures are known for which the ground state is glassy; glasses are

produced by rapid cooling as indicated in Figure 5.12. On the other hand, with

sufficiently rapid cooling, it may be that any collection of atoms forms a glass; a

great variety of metals can form glasses, as discussed by Cargill (1975) or Gilman

(1975),

and so can water, as shown by Debenedetti and Stanley (2003).

Glasses 117

To make a glass, start from a liquid and lower the temperature quickly. Below

the melting temperature

7M,

thermodynamic equilibrium requires atoms to arrange

themselves into a crystal. For fast enough cooling rates, which for window glass

are around 10 K s

, and for nickel are 10

K s

, glass forms instead. Density of

the liquid increases slowly, but viscosity η increases dramatically, up to a value of

around 10

Pa s, obeying the empirical formula, known as the Vogel-Fulcher law

η (X exp [C/(T - T

)} . C is a constant (5.50)

There is now a variety of theories, discussed by Angeli (1988), which can calculate

a divergence similar in shape to that of Eq. (5.50). Gradually, the material changes

its mechanical nature from a viscous liquid to a (frequently) brittle elastic solid.

The first peak in the correlation function g(r) narrows, indicating an increase in

short-range order, and the second peak then splits, as shown for amorphous nickel

in Figure 5.10.

(B) Composition (C) Composition

Figure 5.12. (A) Schematic equilibrium phase diagram for a two-phase system with a

eutectic. Below the temperature

the alloy phase separates into a- and /3-rich regions. If

the system is cooled sufficiently rapidly from the liquid phase, then the boundaries of the

solid-liquid coexistence region apparently collapse together, and metastable phases appear.

In case (B), phases a and β have similar crystal structures, and rapid cooling produces a

continuous solid solution. In case (C), the crystal structures of a and β are incompatible,

and rapid cooling in the central region produces a glass. Single-component glasses fit

roughly within this framework if

one

component is taken to be vacuum. [After Perepezko

and Wilde (1998), p. 1074.]

In addition to these incremental changes, there is a deceptively definite temper-

ature at which the specific heat and thermal expansion coefficient change abruptly,

by a factor of around 2, known as the glass transition

temperature

and indicated in

Figure 5.12 by

TQ.

The glass transition is difficult to define precisely, and many

features of the problem remain controversial, as discussed by Cusack (1987) or

Yonezawa (1991). The precise location of the transition is crucially dependent

upon the amount of time one is willing to spend looking for it.

118 Chapter 5. Beyond Crystals

4r l

180

200 220

Temperature T (K)

240

Figure 5.13. Specific heat cp times thermal conductivity κ for the glassy liquid glycerol as

a function of temperature, at three different frequencies. The thermal conductivity does not

vary rapidly, so the experiment basically measures c

. The top portion of the figure shows

the real part of

cpK,

while the bottom portion shows the degree to which the phase of the

heat oscillations being injected differs from the phase of the temperature oscillations being

measured. Note that there is always a transition in the specific heat, but that the transition

temperature increases roughly as the logarithm of the frequency v, meaning that one can

move the transition temperature down by waiting exponentially longer. [Source: Birge and

Nagel (1985), p. 2675.]

An experiment demonstrating this effect particularly well has been carried out

by Birge and Nagel (1985), illustrated in Figure 5.13. A heat source whose heat

output oscillates in time is placed within a glass, and the temperature variation in

the glass is measured, as a function of the mean temperature and as a function of the

frequency of oscillation v. For temperature oscillations above a certain frequency,

the specific heat is low, because many degrees of freedom are unable to respond

quickly enough to contribute to the specific heat. The specific heat rises rapidly

as the frequency falls, because more modes are able to follow along. The particu-

lar frequency at which the changeover occurs is an exponentially rapidly varying

function of temperature. The picture suggested by this and other experiments is

that if one observes some dramatic change in the behavior of glass as a function

of temperature, it is due to the time scale for some process in the glass passing

rapidly across a threshold of patience. Still, there remains the question in principle

of whether there really is a transition temperature

TQ,

as in Eq. (5.50), at which

viscosity diverges, and a glass reaches some type of ideal glassy state. Menon and

Nagel (1995) provide tentative experimental evidence of a true divergence.

Rapid cooling permits many metals to be formed in a glassy state, and com-

puter experiments on argon and hard sphere systems indicate that these too would

form glasses if only one could cool them fast enough. Materials that form glasses

reluctantly, by virtue of very rapid cooling, are called fragile. By contrast, there is

Glasses

119

a tendency among certain materials to form glasses especially readily. These are

strong or network glasses.

The prototypical network glass is S1O2, which is at the heart of most commer-

cial glasses. The basic structure is that of a silicon atom, which wants to bond

with four neighbors, surrounded by four oxygens, each of which wants to bond

with two neighbors. One model of the glass structure is the continuous random

network, shown in Figure 5.14. It is crucial that the angles between all the bonds

be somewhat variable, or else a crystalline structure must resuk instead.

Figure 5.14. The continuous random network was proposed as a model of S1O2 by

Zachariasen (1932). Each silicon tends to have four neighboring oxygens and each oxygen

tends to have two neighboring silicons. The picture was produced by a molecular dynamics

simulation where silicon and oxygen were raised to a high temperature and then rapidly

cooled and allowed to settle (stereo pair).

A rough argument of Phillips (1982b) gives an idea why mixtures where the

the average coordination number z lies between two and three tend to form glasses.

If one views the solid as a mechanical system where atoms bond to neighbors,

then it tends to form a glass when the number of degrees of freedom in the system

just equals the number of mechanical constraints. If there were more degrees of

freedom than needed to optimize the constraints, then the network would be me-

chanically unstable and would flop around, as for polymers (Section 5.8). If there

were fewer, the local structure of the network would have a deep energy minimum

at some particular configuration, and crystals would be favored. More specifically:

When the average coordination number provides z neighbors per atom in a system

of N atoms, the system has a total of Nz/2 bonds and must try to choose an opti-

mum length for each of them, providing Nz/2 constraints. Angles between bonds

provide more constraints. An atom with two bonds must try to optimize the angle

between them. An atom with three bonds must try to optimize the three angles

between the three pairs. For these two cases, the number of constraints to try to

satisfy is given by N(2z

—

3). Setting the total number of degrees of freedom, 3N,

120 Chapter 5. Beyond Crystals

equal to the total number of constraints,

3Ν = Ν(2ζ-3) + γ, (5.51)

it follows that

z = 2.4. (5.52)

Thus according to this argument, systems form glasses roughly when the average

number of neighbors per atom lies between 2 and 3. This argument is in accord

with the picture of the continuous random network in Figure 5.14 and is consistent

with observations in silicon oxides, boron oxides (B2O3), and the chalcogenide

glasses (AS2S3 and As3Se, for example). It does not fit amorphous silicon, how-

ever, which has fourfold coordination. In attempting more detailed accounts of

structure, the most frequently studied case is S1O2. While the general picture

shown in Figure 5.14 is correct, attempts to calculate the distribution of bond an-

gles and compare predictions with experiment have not yet been conclusive.

5.7 Liquid Crystals

O -N=N O CH

^ ►

Figure

5.15.

Picture of the organic molecule p-azoxyanisole

(PAA),

which forms

nematic

liquid crystal between 116°C and

135

°C.

It can roughly be regarded as a rigid rod of

length 20 Â and width

Â.

Intermediate in order between liquids and crystals are the liquid

crystals.

Their

mechanical properties are those of a liquid, yet certain types of order, particularly

in orientation, persist over large distances. The main structural element is a rodlike

molecule as shown in Figure 5.15, often made by two linked aromatic rings with

various flexible chains hanging off the ends.

5.7.1 Nematics, Cholesterics, and Smectics

Nematics. A nematic liquid crystal consists of a series of rods whose centers

are arrayed randomly, as are molecules in a liquid. Just as in a liquid, there is no

long-range positional order. However, there is long-range orientational

order,

Liquid Crystals 121

shown in Figure 5.16. As a consequence, the refractive index of the liquid varies

by around 20% in different directions and must be regarded as a tensor. This tensor

has complete rotational symmetry about the axis n and has mirror symmetry about

the plane normal to n as well as the planes containing it; its point group is D^/, in

Schönflies notation, or oo/mmm in international notation.

Figure 5.16. The molecules of

nematic liquid crystal have long-range orientational order,

but only short-range positional order.

Cholesterics. A variant of the nematic liquid crystal is the cholesteric (see Figure

5.17). Now the director n rotates slowly along an axis that is perpendicular to it,

described by

= 0 (5.53a)

= cos qox (5.53b)

= sin qox. (5.53c)

The wavelength of the twist λ = Ιπ/qo is on the order of thousands of angstroms

and is therefore much larger than the lengths of

the

molecules. This length can vary

rapidly as a function of temperature. The twist breaks the mirror symmetries about

the planes containing the x axis, and this loss of symmetry is directly connected

to the fact that cholesterics are produced by chimi molecules; these molecules are

rodlike, but they twist slightly as one moves up the rod.

Smectìcs. The final major class of liquid crystals is the smectic. Now, there is not

only long-range orientational order, but long-range positional order in one direction

as well. In smectìcs A, the rodlike molecules arrange themselves in layers with a

well-defined spacing, although within each layer the structure is liquid-like. The

layers are perpendicular to the director n. In smectìcs C, the director is no longer

perpendicular to the layers, and it may or may not rotate as one moves along the x

axis,

depending upon whether or not the liquid crystal is made of chiral molecules.

122

Chapter 5. Beyond Crystals

Figure 5.17. The director

cholestenc liquid crystal rotates as one moves along the x

axis.

Finally, in smectics B, molecules are arranged in a crystalline fashion within the

layers, and only the fact that the layers slide about with respect to one another

distinguishes the structure from a perfect crystal. These three phases are shown in

Figure 5.18.

Figure 5.18. The three main smectic phases, A, B, and C display one-dimensional long-

range positional order.

5.7.2 Liquid Crystal Order Parameter

One way to define an order parameter that will pick out nematic order is to consider

the probability n\ that a particle is at

pointing along θ\ and define

Ό= ί ά

ηαθι

ηι

(7ι, 0i)i(3 cos

0i - 1). (5.54)