Marder M.P. Condensed Matter Physics

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Polymers 123

The use of the quadrupole moment rather than the dipole moment is dictated by

the fact that nematic molecules are unchanged when flipped through 180°, and one

does not want to get into the business of deciding which way they are pointing. If

they are taken to point up and down with equal frequency, then a dipole moment

vanishes. Another possible order parameter is produced by the tensor of dielectric

constants, or of magnetic susceptibility, both of which change their symmetries

when nematic ordering sets in. So that the result vanish in an isotropie phase, one

might define

Qaß =

Caß —

^αβ 2_^ £77, (5.55)

which picks out the anisotropie part of the dielectric tensor. For a computation of

the mechanical properties of nematic liquid crystals, see Section

12.4.1,

and for

more detailed discussions of the topic see Chandrasekhar (1992), de Gennes and

Prost (1993), and Chaikin and Lubensky (1995).

5.8 Polymers

Polymers, like liquid crystals, are built from rod-like molecules, but now the rods

are floppy and exceptionally long. Polyethylene, for example, consists of thou-

sands of repeating units of

CH2.

One repeating unit is a homopolymer, while two

or more in alternation form a copolymer. The degree of polymerization is the num-

ber of basic units repeating in a typical chain. In useful materials, this number may

be in the tens to hundreds of thousands.

It may seem unlikely that any conceptually simple picture could capture fea-

tures of polymer behavior. It is from the enormous lengths of the individual poly-

mer chains that simplifications can flow. The molecules are so long that they be-

have like ideal floppy chains, wiggling randomly in an environment produced

self-

consistently by all the other polymers. The starting point for study of polymers is

therefore a single polymer chain immersed in a solvent liquid.

View the polymer as a collection of identical rigid segments, connected by

joints,

each of which is completely free to rotate as it wishes, depicted in Figure

5.19. Different segments of the chain are even free to rotate right through each

other, an admittedly unrealistic feature of the model that needs to be corrected in a

more sophisticated treatment. It may also seem unrealistic to reduce complicated

bending energies to rigid rods and joints, but this particular simplification makes

little difference.

5.8.1 Ideal Radius of Gyration

One of

the

most important features of an isolated polymer chain is its characteristic

size,

called the radius of gyration Jl, which is the root mean square distance from

one end of the polymer to the other. This quantity is the same as the mean square

distance traveled by a random walker, Section 5.2.4, but it is interesting to derive

it in a different way more directly related to polymer physics. Suppose one end of

124

Chapter 5. Beyond Crystals

Figure 5.19. Illustration of a polymer as a random walk. To prepare this figure, each

segment was permitted to turn at an angle up to 26° relative to the previous segment.

the polymer to be sitting at the origin, and suppose that one has already calculated

the probability 3V(/?') that a polymer of length N has its end at position R'. Then

the probability JV+i (R) that a polymer of length N +

has its end at position R is

N+i

(R)=

[ dR"y

{R')y\{R~R') The ontyway for the end of the polymer (5.56)

J to be at R is for the end of the

/Vth

segment to have been at R', and for the

(JV

+ l)st segment to reach from R' to R.

=4>

N+

i(k)

(Ϊ)7\

(k) This is the convolution theorem for the (5.57)

Fourier transform of

CP;

Section A.5.

=> 7

(%) = [7\

(k)}

By recursion. (5.58)

Provided N is large enough, details of 3Ί (k) will not matter; one only needs to

know its behavior near k = 0, where it can be expanded in a Taylor series. Keeping

terms up to order k

gives (Problem 7)

3>,

(it) «

- |/t

ck2

(5.59)

=> 3V(£) xie~

NCk2

This approximation is excellent for large N. (5.60)

=>■ 7N{R} — e /2A'C Inverting the Fourier transform. (5.61)

VÏ^NC

The statement that the probability of a large number of uncorrelated events is de-

scribed by a Gaussian, as in Eq. (5.61), is known as the central limit theorem. One

can easily calculate the constant c from

Ç= \- iPt(Â:)=û /3 This can be used as a definition of the length (5.62)

ßfc2 £=0

α 0

individual polymer segment.

Polymers

125

and can also relate

it to the

ideal radius

gyration

Uli

the polymer,

dRR

(R)

3CN

= a

write«

+RJ

note that the

(5.63)

J three terms

in the

sum must have

the

same integral,

and

perform

the

integral

for any one

them.

=φ.

3^i = a\N.

This expression

identical

Eq. (5.16) with (5.64)

= t/to, but has

been derived

in a

different

way.

Polymer Interactions. Two features

this calculation appear questionable. First,

the polymer

is not

actually composed

identical segments that pivot completely

independently

one another.

reality, the polymer

rather stiff when viewed

the molecular level. However,

two

stiff springy units placed

end to end

have half

the spring constant

the original unit, four placed together have one-fourth

the

spring constant,

and by the

time

one has a

chain made

up of a

million segments,

one

can

safely view

it as

composed

of, say,

20,000 completely floppy sections,

each built from

50 of the

original springs. Making this idea more precise

is the

subject

Problem

8. The

result

that expressions (5.61)

and

(5.64) survive,

but

the constant

must

defined

in a

more general way.

The more serious defect

the calculation

that

allows different points

the polymer to slide freely through each other. One should recognize that the poly-

mer

can

never visit

the

same point twice during

its

path;

the

polymer describes

self-avoiding random walk (SARW). Such

calculation presents many formal

dif-

ficulties. A considerably simpler method begins

calculating

the

forces needed

to extend

compress

idealized polymer,

and

then

determines whether

the

interactions between polymer segments

are

powerful enough

cause

the

overall

shape

the polymer

alter noticeably from

its

idealized shape.

Stretching

Polymer. From

Eq.

(5.61)

one can

determine

the

force needed

stretch a polymer. Fix one end at the origin, grab the

far

end, and move it to location

This motion costs

energy (because

all

polymer segments rotate freely)

but

reduces

the

entropy.

For

example,

if the end

were

to be

pulled

far

enough that

the whole polymer was made completely straight,

could

longer move,

and its

entropy would

zero. More generally,

the

entropy associated with

any

restricted

configuration is just Boltzmann's constant

times

the log of

the probability that

the configuration will occur. Because Eq. (5.61) gives

the

probability that

the end

of the polymer be found

R, one has immediately that

■2

»2

= So k

y Use

also

Eq.

(5.64) relating

a and X,. S

(5.65)

3^T a

constant independent

of R

whose value

unimportant.

The free energy resulting from this entropy

?0+lk

T^=3

T^-.

Use

Eq.

(5.64).

(

.66)

2 a

126 Chapter

Beyond Crystals

and the force F needed to pull the end of the polymer away from its preferred

location at R = 0 is

F =

T-^

^R = -R. (5.67)

N N

The polymer behaves like an ideal spring of vanishing equilibrium length, with a

spring constant that rises in proportion to temperature, and that falls in proportion

to the molecular weight

3l\

oc yv

of the polymer chains.

Compressing a Polymer. While the force needed to stretch a polymer is well

indicated by grabbing the far ends and pulling outwards, this calculation says noth-

ing about the forces needed to compress a polymer into a small volume, because

not only the far end, but all intermediate points, must now be pressed inwards. A

simple estimate of the forces needed for this compression is obtained by imagining

a polymer molecule trapped within a sphere of diameter Jl <C 3?i. Suppose that

wherever the polymer hits the wall of the sphere it sticks permanently, and guess

that by finding the free energy of a configuration of this type, one obtains a rea-

sonable estimate of the free energy in the more realistic situation where points of

contact between polymer and sphere change with time.

The typical number M of monomers separating points of contact along the

polymer should be

M r^i Since a polymer of length M has characteris- (5.68)

Ü tic size

\/~Ma

So now one has N/M segments of molecular weight M, each stretched out to dis-

tance

01.

According to Eq. (5.66), the free energy of such a collection of polymers

N λ 1ί

^ιΝ ri

The pressure the chamber exerts upon the polymer is therefore

(5.69)

d a

T{N/M)

^ ~ ~ ΛΦ3"

^ö?2

^ ΦΤ— '

Drop constants of order umty

· (5.70)

which is j ust the pressure of an ideal gas of N/M particles in volume Ji

Volume

Interactions. Although the calculations of the forces needed to extend or

compress a polymer will later have direct use in finding the mechanical properties

of polymer mixtures, the goal now is to use them in order to determine the impor-

tance of interactions between portions of the polymer chain which are not directly

adjacent to one another. These interactions are rare when the polymer describes a

random walk within a solvent, and therefore their contribution to the free energy

should be contained within the virial expansion of statistical mechanics. One need

know nothing of the elaborate techniques used to find the coefficients in this ex-

pansion; one simply needs to use the fact that the virial expansion is in powers of

the density of particles.

Polymers

127

Suppose that when the interactions between distant points on the polymer

chain, called volume interactions, are taken into account, the radius of gyration

of the polymer becomes ul, rather than %. The density of particles available to

interact with one another is therefore

/V 'V?

= —- = Factors such as 4π/3 are out of place in a (5.71)

3? ü ÜI qualitative analysis of this type.

According to the virial expansion, interactions between particles should produce

contributions to the free energy of the form

J (X k

T$.

[An + Bn

+ Cn

+ . . .]. The coefficients A, B, and C depend upon (5.72)

temperature. The factor of X

appears

because 3 is extensive.

Adding together Eqs. (5.66), (5.69), and (5.72) therefore gives an estimate of

the free energy of a polymer whether it shrinks or expands, and includes the effect

of interactions between distant segments. Dropping constants of order unity, one

has

+ k

ft?

-\\.f ft? \ / ft? \

„( ft? λ

(5.73)

The condition that 3 have a minimum as a function of

rn <τ>2

<Τ)4 φ6

>-f\- *A.T ^Λ,τ J\.r

2^-2-^

-3ß-

-i-r-6C-^=0. (5.74)

utf

If the second virial coefficient, B, is positive, then polymer segments tend to repel

one another, and the polymer should swell to a larger size than

Ui\.

^> 3£i, then

the only two terms that are significant in Eq. (5.74) are

75)

=*> 3?

OC —~r =>- 3Ì

3/5

. The guess that % » % is confirmed. (5.76)

If on the other hand, the second virial coefficient B is negative, the polymer has

a tendency to shrink. The pressure in Eq. (5.70) resists shrinkage, but it is over-

whelmed by a negative pressure from the virial expansion. The two most important

terms in Eq. (5.74) now give

Any tendency toward attraction of distant polymer segments leads the whole mol-

ecule to collapse into a small ball, whose volume is proportional to the molecular

weight.

Finally, if B = 0 and C is not too large,

is minimized for

= 3tr; the polymers

behave like an ideal random walk. A solvent tuned so that B vanishes is called a

Θ solvent; polymers in θ solvents execute nearly ideal random walks, as in Figure

5.19.

128

Chapter 5. Beyond Crystals

Figure 5.20. Schematic drawing of light path through sample filled with micron-scale

particles in diffusing-wave spectroscopy.

5.9 Colloids and Diffusing-Wave Scattering

5.9.1 Colloids

Many familiar liquids including milk, hand creams, and blood are colloids, de-

scribed by Gast and Rüssel (1998), which means that they are made of particles

whose size ranges from nanometer to a micrometer dispersed in a liquid or gas.

Electrostatic interactions between the particles can keep them from clumping to-

gether, and when their size is comparable to the wavelength of light, even a small

fraction of colloidal particles can render a system opaque, as in the case of clouds

and fog.

While colloidal particles can form crystals, it is more common for them to be

arrayed randomly and move constantly, so that one can only describe their location

in a statistical sense. Their strong interaction with light opens up the possibility

of studying the spatial arrangement. However, neither the theory of weak scatter-

ing from crystals nor dynamic light scattering of Chapter 3 are adequate because

light scattering from colloidal particles is dominated by multiple

scattering.

A typ-

ical photon travels on a convoluted path through many particles before exiting the

liquid.

5.9.2 Diffusing-Wave Spectroscopy

The technique of diffusing-wave spectroscopy, developed by Weitz and Pine (1993)

exploits the multiple scattering limit in order to measure features of the correlation

function Eq. (3.49) that conventional scattering does not. It measures the mean

square distance that particles move in time t:

i n Look back at the definition in Eq. (3.49). In

(Ar

(t)) = - drdr n

(r, f\ t)\f- r'\

Γε8εη

< ef <\

the

most important corre-_

N I ±\ ' ' /\ I lations will be when / = / and particle lo-

cations are correlated with themselves over

time.

(5.78)

How does this correlation function emerge from the process of multiple scat-

tering illustrated in Figure

5.20?

The basic idea is that when light scatters from

Colloids and Diffusing-Wave Scattering 129

many particles, its phase when

exits the sample is determined by the total path

length through which it has traveled. Even small changes in the locations of parti-

cles produce changes in phase. Indeed, multiple scattering increases sensitivity to

particle motions. Dynamic light scattering can detect particle motions on the order

of a wavelength of light, while diffusing-wave spectroscopy is sensitive to much

smaller motions just so long as the cumulative total distance moved by particles in

a light path is comparable to a wavelength.

The starting point is the Siegelt relation

Eq. (3.64) which- shows that the

dynamic correlation function (E(t)*E(0)) can be deduced from measurements over

time of light intensity exiting a sample at a point.

However, when light scatters many times, the relation between particle loca-

tions and (££(0)£

(f)) is no longer given by Eq. (3.59). Instead, specializing to

linear polarization and dropping subscripts on the electric field to simplify matters,

let p denote some path that light takes to get from the laser to the detector. Then at

the detector

E*(0)E(t]

\Eo\

( (Σ E>"

M0)

) (E

Ε,&Α \

(5.79)

Here E

is the amplitude of the electric field along path p and φ

is the phase of the

electric field at the end of path p. Assume next that the strongest correlations come

from electric fields along the same path at different times, and that the intensity I

is not correlated with the phase φ

. If this is correct, then only terms where

= p'

need be kept in Eq. (5.79), and the average of the product can be replaced by the

product of averages giving

E*(0)E(t))

IJL

ί(Φ

{ή-Φρ(0))\ ,.

„ _

.„^

|£ol £—' \

/ \ /in

amplitude of E

over time

t is

not worth

P considering, just the change

phase.

the intensity

(5.80)

Phase. What is the change in phase of light that escapes from the sample? For

the scattering event that takes light from particle r, to particle r

i, the wave vector

of light is

k:(t) = kn—

Since for elastic scattering the wavelength

of (5 81)

i(t)-7i(t)\ light is

fixed

Thus the phase is

(ή-φ

(0)

- Σ Ut)

■

[r,

(i)-r,(0] -*i(0) · [r,-+i(0)

-r,-(0)].

(5.82)

i"=0

As shown in Problem 9, this phase can be rewritten and approximated as

(ή -φ

(0)π-Σ

Âi ■

ΔΓ/(ί) (5.83)

i=l

130

Chapter 5. Beyond Crystals

where hqi is the momentum transfer of the scattering event from particle i

— 1

£,

*,·(())-jfc;_i(0)

and

An{t) = n{t) -r,(0). (5.84)

It is now possible to evaluate contributions from the phase in Eq. (5.80). Write

-ΐ(Φ„(ί)-φ

(0))\

= h

Σ;

?.ΓΔΟ(0\

From

Eq.

.83).

85)

Uj^+i^j-Ar^-^qj-Arjit)]

(5.86)

(

Odd powers of

cjj

should vanish since

(

Jr [fl / ·

jit)}

. . .))

positive and negative values are equally

(5 87)

-li.

■>

'' j I

likely. Furthermore products such as

Ar^Ar

^ should vanish because

motions

of different particles are

uncorrelated.

«Πθ-ϊ^-ΔθίΟ]

))

(5-88)

(

\ The justification for this move is called

<[^·ΔΟ(0]

) ^ffi^Ä

(

)

normally distributed random variables

Triantafyllopoulos

(2003).

There is no reason that scattering paths

q*j

and particle displacements A7j should

be correlated, so one can write

y2((qj

■

A7j(t))

— (q

) (Ar

(t)} See Problem 9. (q

) means choose any q

as (5.90)

their statistical properties are all the

same.

;=i

The average over (g

) could be a terribly complicated formal problem, since q

kj(0) —kj-\(0) varies from 0 to

2ko

depending on the direction light scatters from

particle

—

1. The standard approach to this problem is to evade it by defining

)^2k

(5.91)

Here / is the mean distance between scattering sites, and

can be interpreted as the

mean free path of light in the medium. When /* is very large, kj is nearly parallel

to kj-\ and (q

) is small. When light leaves each particle in

nearly random

direction, /*

/. Using this definition, one has finally

-iï>„(r)-<M0))\

Σ; ?;-ΔΓ;(/)\

,-^(Ar

(/))/(3/*)

92)

where s = NI is the total path length of light traveling through the sample.

Colloids and Diffusing-Wave Scattering 131

Amplitude. To evaluate Eq. (5.80) still requires dealing with the light intensity

Ip of path p. The approach to this problem is to make further use of the idea that

light is executing a random walk through the colloidal system. Suppose a flash of

light hits the sample at time 0, and exits it at time t. One can immediately deduce

that the path length of the light in the sample was s = ct. This means one can write

Eq. (5.80) as

77,/^im^^w

93)

|£br J

;

The factor of /* keeps dimensions correct. All the information about the probability of

having a path of length s is contained in the solution of the diffusion equation that follows.

Since the probability distribution of particles undergoing random walks obeys

the diffusion equation (Section 5.2.4), the intensity of light leaving the sample

should be the solution of the diffusion equation, which means a solution of

dU,

-^- =

T>,W

(5.94)

where

\J\

is the energy density of light, and D/ is a diffusion constant for light. To

estimate D/, imagine that the light path is a random walk with step length conven-

tionally given as

21*.

Then according to Eq. (5.14) the diffusion constant is given

T>i

cl*/3.

(5.95)

Boundary conditions for the diffusion equation are tricky, and there seems

some temptation to decide between mathematical boundary conditions on the grounds

of agreement between theory and experiment. The problem is that the actual

boundary conditions follow from the fact that once the source turns off, all sub-

sequent light waves are outgoing waves at the boundary of the sample. However,

approximating light as a diffusing field allows no way to impose this condition ex-

actly. The issue is discussed in most depth in Chapter 9 of Ishimaru (1978). For the

purposes of discussion here, a particularly simple boundary condition will be em-

ployed, which is that the energy density of diffusing light £// vanishes at the sample

boundaries, because photons that arrive there are immediately whisked away as

propagating radiation.

A model calculation that corresponds reasonably well to experiments is to say

that at time t = 0 a narrow plane wave of light of intensity

Ui(x, 0) = ΙοΙ*δ(χ — Χο)/8π The multiplicative factors have been chosen (5.96)

to give correct dimensions.

is present in a sample at depth

> 0. The sample is a slab extending from 0 to

L in the x direction, and off to infinity in the other directions. Given this initial

condition, the diffusion equation (5.94) can be solved by Laplace transforms. Let

poo

Ü(r, a) = /

dt U(r, t)e~

. (5.97)

132 Chapter 5. Beyond Crystals

-/ _

Figure 5.21. Intensity of light scattered back from a suspension of polystyrene spheres in

water, volume fraction 5%. The variables displayed in the plot should be linearly related

according to Eq. (5.104) and they are. [Source: Weitz and Pine (1993), p. 677].

Then

αυ(χ,α)=1*Ι

δ(χ-Χο)/(8π) + Τ>ι^υ(χ,α) Only thexcomponent of r (5.98)

matters because this is a plane

problem.

Ü(x, a)=A cosh

+ B sinhxK+ . , J?

-l*-*ol«

IÓTTU/K

where

(5.99)

(5.100)

How can one relate the energy density of diffusing light to the intensity / ar-

riving at a photodetector? The energy current arriving at the edge of the sample

is D/Vi/. Set this equal to the electromagnetic energy current density cE

/8n =

cl/ΰπ, obtaining

E*(0)E(t]

From Eqs. (5.93). Use the energy current D/Vi/ to find (/). Evaluate dU/dx at the point

where light is to be collected.

[ ds 8π {Vi/I

cl*)^-U{x, s/c)e-

{Ar2{t))/{3n

(5A0l)

(SKD/I

1*)—U(X,

a), where a = ckl(Ar

(t))/{3l*

(5.102)

In a forward scattering geometry, light is collected at the far end of the sample,

x = L. Imposing the boundary conditions t/(0, a) = Ü(L, a) = 0 and solving for