Lecture Notes in Polymer Physics

A. C. Shi

2011.1.12 ~ 2011.1.14

Noted by Y. X. Liu with additional derivation details

1. One mode approximation

Assume the function to be approximated can be expanded in a series

( ) ( ) nik xn

n

x x e (1)

where n=0, ±1, ±2, ⋯. For one mode approximation, only the first three terms are taken, that is

0 1 10 1 1( ) ( ) ( ) ( )ik x ik x ik xx x e x e x e (2)

In two and higher dimensional space, variables, x and k, become vectors x and k. To

approximate high dimensional functions, one should consider the origin and its neighbors. Using

two-dimensional hexagonal lattice as an example, one mode approximation retains only the origin

term and the six nearest neighbors, depicted in Figure 1.

Figure 1

One mode approximation is not a good approximation for square wave. To expand square

wave function according to eq. 1, n decrease with n in an order of 1/n, which is very slow and a

large number of terms are required to approximate it well enough. This also explains that spectral

method is not suitable to deal with strong segregation systems where the density profile resembles

a square wave function.

2. Spinodal decomposition and nucleation (For more details, see Chaikin and Lubensky,

Principles of Condensed Matter Physics, 1995, p479-491)

Consider the following dynamic equation,

2 FMt

(3)

(It is an equation that describes the evolution of conserved order parameter). In linear instability

analysis, the order parameter is first re-expressed as

0 (4)

where 0 is the average order parameter, and describes the fluctuation. Then eq. 3 finally

reaches the form no matter what the exact expression of F,

( )Pt

(5)

If P()>0, fluctuation amplifies with time because

( )0P te (6)

is an increasing function. In other words, fluctuation grows spontaneously when P() is positive.

This kind of evolution path is called spinodal decomposition. When P() is negative, fluctuation is

suppressed. To form a new phase, an energy barrier should be overcome. This kind of evolution

path is called nucleation.

(a)

(b)

Figure 2

The spinodal curve is, strictly speaking, a mean-field concept. In real systems, where

fluctuations are important, the boundary separating nucleation from spinodal decomposition is not

perfectly sharp. When the free energy barrier is close to or smaller than the thermal energy, the

fluctuation is important. In this case, nucleation and spinodal decomposition behave similarly and

cannot be distinguished from experiments. Therefore, it is impossible to determine spinodal curve

exactly via experiments.

3. Conserved and non-conserved order parameter

Dynamic equation for non-conserved order parameter (NCOP) is

F

t (7)

This is an example of dissipative equation. The degree of orientation for liquid crystal is an

example for non-conserved order parameter.

An intuitive derivation of the dynamic equation for conserved order parameter (COP) is given

below. Each conserved parameter is associated with a continuity equation,

0t

J (8)

where the first term stands for the variation of the order parameter itself, and the second term

represents the change of order parameter due to the in and out flux. In fluid dynamics, it can be

written as

0t

(9)

The flux J is just the gradient of density. In phase separation systems, the expression for J is

MJ (10)

where M is the mobility and is chemical potential. Substitute eq. 10 into eq. 8, and using the

definition of chemical potential

F

(11)

the dynamic equation can be finally written as

2 FMt

(12)

NCOP and COP behave differently in linear stability analysis. For COP, one specific mode will be

selected to grow new phase, i.e. a structure with a certain characteristic length will be formed.

4. Ginzburg criteria (For more details, see Chaikin and Lubensky, Principles of Condensed

Matter Physics, 1995, p214-216)

Ginzburg criteria are the criteria for determining whether or not the mean-field approximation

breakdowns. It quantitatively measures the importance of fluctuations by considering the average

over a coherence volume, V=d of the deviation (x)=(x)-<>, of the local order parameter

from its equilibrium value:

1coh ( )d

VV d x x (13)

Fluctuations are negligible if <(coh)2> is much less than <2

>, otherwise, fluctuations are

important and the mean-field approximation will fail.

Under what conditions will the mean-filed approximation be valid?

(1) High dimension space, d≥4.

(2) Long range interactions.

5. Scattering (For more details, see Chaikin and Lubensky, Principles of Condensed Matter

Physics, 1995, p29-37, p47-49)

In scattering theory, the scattering cross-section 2d

d, which is the differential cross-section

per unit solid angle, is a key quantity. It represents a static cross-section obtained experimentally

by integrating over all possible energy transfers to the medium. In practice, this integration is

naturally accomplished by X-ray diffraction but not by neutron diffraction. In quantum mechanics,

the scattering cross-section relates to the transition rate Mk,k’ in

2 2

,2dM

d k k' (14)

If the scattered particle interacts with the scattering medium via a potential U (and the interaction

is sufficiently weak that only lowest order scattering need be considered for the entire sample),

then by Fermi’s golden rule, the transition rate between the incident (incoming) and final

(outgoing) plane wave states of the scattered particle, |k> and |k’>, is proportional to the square of

the matrix element,

, ( )d i iM U d xe U ek x k' xk k' k k' x (15)

where U(x) is the sum of terms arising from each of the individual atoms in multiparticle system:

( ) ( )U Ux x x (16)

where x is the position of the atom arbitrarily labeled . The matrix element in the scattering

cross-section then has the form

, ( )d i iM d xe U ek x k' xk k' x x (17)

This can be placed in a more convenient form by taking R=x-x:

( ) ( ),

( ) ( )

( )

( )

( )

( )

i id

i id

i id

i

M d R e U e

e d R e U

e d R U e

U e

k x R k' x Rk k'

k k' x k k' R

q x q R

q x

R

R

R

q

(18)

Here q is the scattering wave vector. The relation of q, k and k’ is illustrated in Figure 3. U(q) is

Figure 3

the atomic form factor or Fourier transform of the atomic potential. The scattering cross-section is

'

2*

', '

2( ) ( ) i id

U U e ed

q x q xq q (19)

Eq. 19 expresses the scattering cross-section for a particular configuration, specified by the

position vector x, of atoms in the sample. If the positions of the atoms are rigidly fixed, as they

would be in a classical system at absolute zero, then Eqs 19 and 14 correctly give the cross-section.

In real materials, particles move about, probing large regions of phase space determined by the

rules of statistical mechanics, and some ensemble average of the ideal cross-section is required.

Assuming that time averaging and averages over all allowed configurations (ensemble averages –

denotes by angular brackets < >) are equivalent (i.e. that the system is ergodic) we have the static

or quasi-elastic limit. In this limit, the scattering cross-section is

'

2*

', '

2( ) ( ) i id

U U e ed

q x q xq q (20)

If the atoms are identical, then the form factor in eq. 20 comes outside the sum and the bracket,

and the scattering cross-section from a statistical system becomes

2

2( ) ( )

dU I

dq q (21)

where the function

'( )

, '

( ) iI e q x xq (22)

depends only on the positions of the atoms in the scattering medium and not on the nature of the

interaction between atoms and the scattering probe. I(q) is thus called the structure function.

The structure function can be expressed in terms of density operator and correlation function

as shown below. The number density operator specifying the number of particles per unit volume

at position x in space is defined as

( ) ( )n x x x (23)

where x is the dynamical variable specifying the position of particle . The ensemble average of

the density operator is the average density at x:

( ) ( )n x x x (24)

In homogeneous, isotropic fluids, <n(x)> is independent of x and is simply the average density

n=N/V. In crystals, <n(x)> becomes a periodic function of x. The Fourier transform of the density

operator is

( ) ( )

( )

( )

d i

d i

d i

i

n d xe n

d xe

d xe

e

q x

q x

q x

q x

q x

x x

x x (25)

Here we use the sifting property of the delta function which will be used over and over again

throughout this note:

0 0( ) ( ) ( )dxf x x x f x (26)

We can now express the structure function in terms of density operator:

'

'

( )

, '

'

( )

( )

( ) ( )

i

i i

i i

I e

e e

e e

n n

q x x

q x q x

q x q x

q

q q

(27)

This is simply a Fourier transform of the two-point density-density correlation function, which is

defined as

1 2 1 2( , ) ( ) ( )nnC n nx x x x (28)

It can be seen from

1 2

1 2

1 2

1 2

1 2

1

( )1 2 1 2

( )1 2 1 2

1 1 2 2

1 1 2 2

1 1 2 2

1 1

( , )

( ) ( )

( ) ( )

( ) ( )

( ) ( )

( )

id dnn

id d

i id d

i id d

i id d

i id

d x d x e C

d x d x e n n

d x e n d x e n

d x e n d x e

d x e n d x e

d x e n e

q x x

q x x

q x q x

q x q x

q x q x

q x q x

x x

x x

x x

x x x

x x x

x

1

1

1 1

1 1

( )

( )

( )

( ) ( )

( )

i id

i id

i i

i i

e d x e

e d x e

e e

e e

n n

I

q x q x

q x q x

q x q x

q x q x

x x

x x

q q

q (29)

In a periodic solid, the ensemble average of density operator can be decomposed into Fourier

components with wave vectors in the reciprocal lattice:

( ) in n e G xGG

x (30)

Thus, the average number density in a periodic solid is fully specified by its Fourier components

<nG> at reciprocal lattice vectors G.

Figure 4

If scatterers are rigidly fixed at sites on a periodic lattice, the scattering matrix element (eq.

18) becomes

, ,M V Uk k' G q GG

(31)

This can be seen from

,

( )

,

( )

( )

0, if

( ) , if

i

i i

i

M U e

U e e

U e

V U

q xk k'

G x G q x

G

G x

G

G q GG

q

q

q G

G q G

(32)

The scattering cross-section then becomes

2

22,

dV U

d G q GG

(33)

Thus, there will be peaks in the scattering pattern at every reciprocal lattice vector with intensity

proportional to the square of the volume of the sample and to the square of the Fourier component

of the scattering potential at wave vector G. These are the Bragg scattering peaks of the solid. The

scattering into Bragg peaks is elastic so that the magnitude of the incident and scattered wave

vectors is the same, i.e.,

2 2

'k k (34)

This leads to a variation of Bragg’s law, known as the Laue condition. We have the relation

'q k k (35)

At the Bragg scattering peaks, we also have

q G (36)

Combining eq. 35 and eq. 36 arrive at

'k k G (37)

Squaring the left and right hand side of eq. 37, we have

2 2 2

' 2k k G k G (38)

Using eq. 34 and rearranging eq. 38, we have

2

2 2

G Gk (39)

As shown in Figure 4, the right hand of above equation can be evaluated as

sin2 2

G Gk k (40)

Now eq. 39 can be simplified to be

2 sinG k (41)

This relation is equivalent to the Bragg condition as can be seen by substituting

2

dG (42)

and

2

k (43)

into eq. 41:

2 sind (44)

where d is the distance between adjacent planes and is the wave length of incident wave.

6. Excluded volume and incompressibility

The interaction between hard core A and B is

A B' ( ) ( ') ( ')d dU d rd r Vr r r r (45)

For Flory-Huggins interactions, V has the form

( ') ( ')V r r r r (46)

where is Flory-Huggins interaction parameter. This leads to the well-known interaction energy

A B

A B

A B

A B

' ( ) ( ') ( ')

' ( ) ( ') ( ')

( ) ' ( ') ( ')

( ) ( )

d d

d d

d d

d

U d rd r V

d rd r

d r d r

d r

r r r r

r r r r

r r r r

r r

(47)

In general, V includes the information of excluded volume effect. However, the approximation of

eq. 46 ignores the exclusive hard core interactions (any interaction is set to 0 when A and B are not

in identical location), which can be seen in Figure 5. To avoid unphysical results from this

(a)

(b)

Figure 5

approximation, the incompressibility condition is introduced.

7. Structure factor for single ideal Gaussian chain and Debye function (For more

details, see Chaikin and Lubensky, Principles of Condensed Matter Physics, 1995, p32-33; M.

Doi, Introduction to Polymer Physics, 1996, p9-10.)

Origin of the term structure factor. In section 5, we introduce the structure function in eq.

22. The bracket denotes averaging over all possible configurations. For a system of N atoms, I(q)

contains a sum of N2 complex numbers with phases determined by the positions of all N particles.

If the relative positions of the atoms are random (as for an ideal gas) then the only terms that do

not average to zero are those with a = a’ for which , '

. In this case, I(q) increases

linearly with N (rather than with N2), i.e., I(q) is extensive. For fluid phases, where relative

positions are not random for some close neighbor particles, I(q) remains extensive. An intensive

version of the structure function (independent of N) is obtained by dividing I(q) by N or V. The

resulting function,

1( ) ( )S N Iq q (48)

or

1( ) ( )S V Iq q (49)

is called the structure factor. Most experimental data are presented in the first (dimensionless)

form.

Figure 6

The structure factor of a single ideal Gaussian chain. The Gaussian chain has N segments

which are indexed from 0 to N-1. As illustrated in Figure 6, n and m are independent indexes.

Therefore, the structure factor can be calculated as

( )

,

( )

1( )

1

n m

n m

i

n m

i

n m

S eN

eN

q R R

q R R

q (50)

the ensemble average according to Gaussian distribution:

2

2

( )

33

223

2

3

2

n mi

i

r

n m bi

e

e

d re en m b

q R R

q r

q r

(51)

Using the following relations

( , , )x y zr (52)

2 2 2 2r x y z (53)

x y zq x q y q zq r (54)

the above integral can be split into three parts:

2

2

2

2

2

2

( )

13

22

2

13

22

2

13

22

2

3

2

3

2

3

2

n m

x

y

z

i

x

n m biq x

y

iq y n m b

z

n m biq z

x y z

e

dxe en m b

dye en m b

dze en m b

I I I

q R R

(55)

The first integral is calculated as

2

2

2

2

2 2

2 22 2

2 22

33

22

2

33

22

2

32 2

32 ( )2

2 4

32 ( )2

2

3

2

3

2

x

x

x

x x

x

x

n m biq xx

xiq x

n m b

x iq x

iq iqx

iqx

I dxe en m b

dxen m b

dxe

dxe

dxe

2

2

2

2 22

2

2

2

2

22

( )

4

3( )2 2

4

1 12 2 2

42

4

6

x

x

x

x

x

iq

iq

t

q

q

qn m b

e dte

e

e

e (56)

Here we use the Gaussian integral

2 2

2tdte (57)

Iy and Iz are similarly obtained:

22

6yq n m b

yI e (58)

22

6zq n m b

zI e (59)

Substitute eq. 56, 58 and 59 into 55, we have

22

( ) 6n m

qn m bie eq R R (60)

where

2 2 2 2x y zq q q q (61)

Substitute eq. 60 into eq. 50, and let N→∞,

22

22

2 2

60 0

2 1 16

0 01 1

0 0

1( )

g

qN N n m b

qn m Nb

q R n m

S dn dm eN

Ndn dm e

N

N dn dm e

q

(62)

In last line, n and m are rescaled in the range of 0 to 1, and the radius of gyration Rg2=Nb2/6.

2 2

2 2 2 2

2 2 2 2 2 2 2 2

2 2 2 2 2 2 2 2 2 2 2 2

2

1

01( ) ( )

01

0

0 1

2 2 2 2

2

1 1

1

g

g g

g g g g

g g g g g g

q R n m

n q R n m q R m n

nnq R n q R m q R n q R m

n

q R n q R n q R q R n q R q R n

g g

x n x

dm e

dm e dm e

e dm e e dm e

e e e e e eq R q R

e ex

2 2 2 2

2 2

22 2

2

( 1)2 2 2 2

2 2 2

11

1 1 1 1

2 1

n x n x x n

x n x n

xx n x n

e e ex

e ex x x x

ee e

x x x

(63)

where

gx qR (64)

Substitute above result into eq. 62, we have

22 2

22 2

2

22

22 2

1

2 2 20

1 1 1

2 2 20 0 0

2

12

2 2 0

12

2 2 0

2 4 4

2 1( )

2 1

2

1 1( )

1( )

2 11 1

2

xx n x n

xx n x n

x n

xx n

xx x

x

eS N dn e e

x x x

eN dn N dn e N dn e

x x x

Nx

N d x n ex xe

N d x n ex x

eN N e N ex x xe

N

q

2

2

4 4 2

24

2 2

21

( )

x

x x x

N e xx

Ng x

(65)

The Debye function. The Debye function is

2 2

4

2( ) 1xg x e x

x (66)

Figure 7

Debye function has two asymptotic forms. When x→∞,

2

2( )g x

x (67)

2 2

2( )

g

NS

q Rq (68)

And when x→0,

4

2 24

2( ) 1 1 1

2

xg x x x

x (69)

( )S Nq (70)

8. Response function and fluctuation-dissipation theorem (For more details, see M.

Doi, Introduction to Polymer Physics, 1996, p29-32; Chaikin and Lubensky, Principles of

Condensed Matter Physics, 1995, p35.)

In statistical mechanics, the probability density function is

0UP e (71)

The ensemble average of any physical quantity (take density as an example) can be evaluated

0 0

0 0

U U

U U

Tr e D e

Tr e D e (72)

Here Tr represents trace, which denotes integrating over all possible configurations. The external

potential is

3 ( ) ( )extU d r Ur r (73)

Under the external potential, the ensemble average of density will deviate from <>:

0

0

( )

( )

ext

ext

U U

U U U

D e

D e (74)

If the external field is small, the deviation

( )U

r (75)

can be written as a linear function of the external potential:

3( ) ' ( , ') ( ')extd r Ur r r r (76)

where (r,r’) is called the response function. It can be calculated

( )

( , ')( ')extU

rr r

r (77)

It is related to the Ursell function Snn(r,r’) as follows:

( , ') ( , ')nnSr r r r (78)

The above equation is the fluctuation-dissipation theorem. Below we will give an illustrative

proof.

The Ursell function can be derived from two-point density-density operator or density

fluctuation as follows:

( , ') ( , ') ( ) ( ')

( ) ( ') ( ) ( ')

( ) ( ') ( ) ( ') ( ) ( ') ( ') ( )

( ) ( ) ( ') ( ')

( ) ( ')

nn nnS Cr r r r r r

r r r r

r r r r r r r r

r r r r

r r

(79)

The ensemble average of density under external potential can be rewritten in the term of

equilibrium ensemble average as

0

0

0 0

0 0

( )

( )

ext

ext

ext

ext

ext

ext

U U

U U U

U U U

U U U

U

U

D e

D e

D e e D e

D e D e e

e

e

(80)

Under weak external field, i.e. in the limit

0extU (81)

the exponential terms can be expanded to the first order term,

1

1

1

1

ext

ext

U

U U

ext

ext

ext

ext

ext ext

ext ext

e

e

U

U

U

U

U U

U U

(82)

where the second order term of <Uext>2 is ignored. From above equation, the density deviation is

3 3

3 3

3 3

3

( )

( ) ' ( ') ( ') ( ) ' ( ') ( ')

' ( ) ( ') ( ') ( ) ' ( ') ( ')

' ( ) ( ') ( ') ' ( ) ( ') ( ')

' ( ) ( ') ( ) ( ') ( ')

ext extU U

d r U d r U

d r U d r U

d r U d r U

d r U

r

r r r r r r

r r r r r r

r r r r r r

r r r r r

(83)

Compare eq. 83 with eq. 76, we find the response function

( , ') ( ) ( ') ( ) ( ')r r r r r r (84)

Compare eq. 84 with eq. 79, the term inside the square bracket in eq. 84 is just Ursell function.

Therefore, the proof is completed.

9. Random phase approximation (RPA) (For more details, see M. Doi, Introduction to

Polymer Physics, 1996, p29-35; Chaikin and Lubensky, Principles of Condensed Matter

Physics, 1995, p38-39.)

Let us consider a mixture of two polymers A and B, having degrees of polymerization NA and

NB, respectively. Let A, B be the overall volume fractions of each type of segment, with

A B 1 (85)

The density operators for A and B also satisfy the relation for incompressible system:

A B( ) ( ) 1r r (86)

Letting <…> denote an equilibrium ensemble average, then

A A( )r (87)

B B( )r (88)

The deviation of the segmental density is defined by

A A A( ) ( )r r (89)

B B B( ) ( )r r (90)

From this definition, the following relation is held

A B A A B B

A B A B

( ) ( ) ( ) ( )

( ) ( )

1 1

0

r r r r

r r (91)

So

A B( ) ( )r r (92)

The fluctuation is characterized by the Ursell functions of A(r) and B(r) as follows:

AA A A( , ') ( ) ( ')S r r r r (93)

AB A B( , ') ( ) ( ')S r r r r (94)

BA B A( , ') ( ) ( ')S r r r r (95)

BB B B( , ') ( ) ( ')S r r r r (96)

Using eq. 92, we have

AA BB AB BAS S S S (97)

Let us consider weak external potentials uA(r), uB(r) which act respectively on the segments

of A and B polymers. The change in the system’s potential energy is

3A A B B( ) ( ) ( ) ( )extU d r u ur r r r (98)

Then the linear response theory implies

3( ) ' ( , ') ( ')d r ur r r r (99)

where subscripts and run over A and B. Applying fluctuation-dissipation theorem, we have

( , ') ( , ')Sr r r r (100)

Homogeneous mixture of A and B polymers is fluid which is spatially homogeneous and

rotationally isotropic. This means that the average environment of any point in a fluid is identical

to that any other point and independent of direction. Thus the average properties of a fluid are

invariant with respect to spatially uniform translations through any vector R and with respect to

arbitrary rotations about any axis. Translational invariance implies

( , ') ( , ' )S Sr r r R r R (101)

In particular, we can choose R to be equal to –r’ so that

( , ') ( ',0) ( ')S S Sr r r r r r (102)

depends only on r-r’. Similarly, other spatial quantities also depend on r-r’, such as

( , ') ( ')r r r r (103)

In this form of Ursell function and response function, eq. 76 is a convolution

3( ) ' ( ') ( ')d r S ur r r r (104)

which is most convenient in Fourier space. Using Fourier transform to eq. 104, the left hand side

is

3 ( ) ( )id r e q rr q (105)

and the right hand side is

3 3

3 3

3 3

3 3 ( ' )

3 ' 3

3 '

3

' ( ') ( ')

' ( ') ( ')

' ( ') ( ')

' ( ') ( )

' ( ') ( )

' ( ') ( )

( ) ' ( '

i

i

i

i

i i

i

d r d r S u e

d r d r S u e

d r u d rS e

d r u d RS e

d r u e d RS e

d r u e S

S d r u

q r

q r

q r

q r R

q r q R

q r

r r r

r r r

r r r

r R

r R

r q

q r ')

( ) ( )

ie

S u

q r

q q

(106)

Thus, in Fourier space, the convolution is just a multiplication

( ) ( ) ( )S uq q q (107)

In the above context, we first outline the basic RPA procedure as follows. In general, if an

external filed u(q) is applied to the system, the resulting change in the concentration is given by

(0)S u (108)

Hereafter (q) is dropped from all related physical quantities for simplicity. Now, in reality there

are interactions between components, which we will take into account through the mean-field

approximation. The interaction u changes to an effective interaction

effu u w (109)

where is different from the one in eq. 108, it satisfies

(0)

(0)

effS u

S u w (110)

Re-express above equation in the form

RPAS u (111)

We finally obtain the Ursell function under external potential ueff:

(0)

(0)1RPA SS

S w (112)

The Ursell function for the mixture of A and B polymers can be calculated by following the

RPA procedure. First of all, let us consider the case where the polymers A and B are placed on the

lattice at random, without excluded volume effects or interaction energies. In this case,

(0)AB A B( ') ( ) ( ') 0S r r r r (113)

(0)BA B A( , ') ( ) ( ') 0S r r r r (114)

but

(0)AA A A( ') ( ) ( ') 0S r r r r (115)

(0)BB B B( ') ( ) ( ') 0S r r r r (116)

since the segments of the polymers are linked together. According to linear response theory, the

change of concentration is given by (using eq. 99)

(0) (0)3 3A AA A AB B

(0)3AA A

( ) ' ( ') ( ') ' ( ') ( ')

' ( ') ( ')

d r S u d r S u

d r S u

r r r r r r r

r r r (117)

In Fourier space,

(0)A AA AS u (118)

Similarly,

(0)

B BB BS u (119)

Now, in reality there are interactions between the chains. Through the mean-filed approximation,

the molecular fields acting on the segments are given by

A AA A AB B( ) ( ) ( )w zr r r (120)

B BA A BB B( ) ( ) ( )w zr r r (121)

Further, there is the conservation of volume condition (impressible condition eq. 86), which can be

represented in the following potential form (which can also be viewed as a Lagrange multiplier):

3A B( ) ( ) ( )exclU d rV r r r (122)

Here V(r) is a potential determined from the volume conservation condition. The mean fields

acting on segments A and B are, respectively, wA+V and wB+V, and so the changes of

concentration are given by the following (in Fourier space):

(0)

A AA A AS u w V (123)

(0)

B BB B BS u w V (124)

where wA and wB are Fourier transform of wA(r) and wB(r), which can be calculated as

3 3A AA A AB B

AA A AB B

( ) ( )i iw z d r e d r e

z

q r q rr r (125)

3 3B BA A BB B

BA A BB B

( ) ( )i iw z d r e d r e

z

q r q rr r (126)

where we use the relation

3A A

3A A

3A

( )

( )

( )

i

i

i

d r e

d r e

d r e

q r

q r

q r

r

r

r

(127)

3B B

3B B

3B

( )

( )

( )

i

i

i

d r e

d r e

d r e

q r

q r

q r

r

r

r

(128)

Substitute eq. 125 and 126 into eq. 123 and 124, we have

(0)

A AA A AA A AB BS u z V (129)

(0)

B BB B BA A BB BS u z V (130)

Moving the term before the square brackets in the above two equations to the left hand, and then

subtracting the first equation from the second equation, we have

A BA B AA BA A AB BB B(0) (0)

AA BB

u u zS S

(131)

With the following relation

A B (132)

AB BA (133)

we can rewrite eq. 131 to be

A A B AA BB AB A(0) (0)AA BB

1 1 12u u z

S S (134)

Further rearrangement leads to

1

A A B(0) (0)AA BB

1 12 u u

S S (135)

where is the Flory-Huggins interaction parameter which is defined as

AA BBAB2B

z

k T (136)

From eq. 107 and eq. 135 the Fourier transform of the concentration fluctuations is given as

follows:

(0) (0)AA BB

1 1 12

( ) ( ) ( )S S Sq q q (137)

The above approximation has used the completely random state as a base, and has estimated

the effect of interactions through a perturbation calculation. Therefore, this model is not applicable

to systems with strong correlation effects, for example a solution near c* where there are large

fluctuations in the concentration. However, the accuracy of this approximation improves as the

concentration increases, and it holds quite well for polymer blends.

10. Landau theory and phase transition

See Chaikin and Lubensky, Principles of Condensed Matter Physics, 1995, p151-188.

11. Variational mean-field theory (For more details, see Chaikin and Lubensky, Principles

of Condensed Matter Physics, 1995, p198-201, p.204-205)

Variational mean-field is a mean-field theory valid for all ranges of temperatures for systems

with order parameters of essentially arbitrary complexity. This variational method is based upon

approximating the total equilibrium density matrix by a product of local site or particle density

matrices and if often referred to as Trln mean-field theory.

Let be any random variable, which can be either continuous or discrete, and let

( ) 0P (138)

be its associated probability distribution. Then the expectation value of any function f() is

( ) Tr ( ) ( )f P f (139)

where Tr signifies a sum or integral over all possible values of . The inequality

e e (140)

valid for any probability distribution, may be proved as follows. Consider the following function

( ) 1F e (141)

Its first-order derivative is

'( ) 1F e (142)

When =0,

(0) 0F (143)

When >0,

'( ) 0F (144)

i.e. F() is an increasing function. Thus, F()>0. When <0,

'( ) 0F (145)

i.e. F() is a decreasing function. Thus, F()>0. In summary, for all F()>0So we have the

inequality

1e (146)

Therefore,

1

e e e

e (147)

which, when averaged over P(), implies

1

1

1

1

e e

e

e

e

e

(148)

which establishes eq. 140.

Now consider a classical Hamiltonian H that is a function of a discrete or continuous classical

field . Let () be any classical probability distribution satisfying

Tr 1 (149)

and

( ) 0 (150)

The canonical partition function can be written as

ln

ln

ln

Tr

1Tr

Tr

Tr

H

H

H

H

H

F

Z e

e

e e

e

e

e

(151)

where < > signifies an average with respect to the density matrix and where F is the free energy

(or more precisely the thermodynamic potential) associated with H. Thus, using eq. 151 and the

inequality in eq. 140, we obtain

ln

ln

HF

H

e e

e

(152)

or

ln

Tr Tr lnB

B

F F

H k T

H k T

(153)

where F is an approximating free energy associated with the density matrix . is a minimum with

respect to variations in r subject to the constraint

Tr 1 (154)

when

1 HeZ

(155)

is the actual equilibrium density matrix. This can be seen from the equation

(ln 1)B

FH k T (156)

where is a Lagrange multiplier whose value is chosen to impose the constraint eq. 154. Thus, at

its minimum with respect to is the actual free energy F.

* Equilibrium state: the system has a free energy that corresponds to the maximum number of

possible configurations. Note that each configuration has the same probability to appear. To

simply understand this, use a two dices experiment as an example. Sum of the points of these two

dices free energy. A certain combination of two points on these two dices configuration. The

equilibrium state when the sum = 7: it has six configurations {1,6}{6,1}{2,5}{5,2}{3,4}{4,3}.

The mean-field approximation. The inequality, Eq. 153, provides the basis for variational

approximations to the free energy that can be implemented as follows: a functional form with free

unspecified parameters is chosen for a trial density matrix to approximate the actual density

matrix. The trial density matrix with the chosen functional form that best approximates the actual

density matrix is obtained by minimizing the approximate free energy F with respect to the free

parameters in . Mean-field theory is obtained by a trial density matrix that is a product of

independent single matrices. If is the single particle density matrix depending only on the

degree of freedom of particle , the mean-field density matrix is

(157)

and the variational mean-field free energy is

Tr lnBF H k T (158)

The precise form of <H> will, of course, depend on H. The second term in eq. 158 is obtained as

follows:

1 2 3 1 2 3

1 2 3 1 1 2 3 2 1 2 3 3

2 3 1 1

1 3 2 2

1 2 4 3 3

1

ln Tr ln

Tr ln

Tr ln

Tr ln ln ln

Tr ln ln ln

Tr ln

Tr ln

Tr ln

Tr ln 1 2 2 3 3Tr ln Tr ln

Tr ln

(159)

We can connect the single particle density matrix with density operator by

Tr (160)

Example: derivation of Flory-Huggins theory for mixture of small molecules A and B. In

the mixture, there are NA molecules A and NB molecules B. Let xA denote the positions of

components A and xB denote the positions of components B, where and are indexes varies

from 0 to NA-1 and 0 to NB-1, respectively. The density operators are

( ) ( )n x x x (161)

where runs over A and B. With the mean-field density matrix is

A A B B( ) ( )x x (162)

the average number density of molecules A can be calculated according to eq. 160:

A A

A

'A A B B A

'

'A A B B A

'

'A A A B B

'

'A A A

'

( ) ( )

Tr ( )

Tr ( ) ( ) ( )

Tr ( ) ( ) ( )

Tr ( ) ( ) Tr ( )

Tr ( ) ( )

n x x x

x x

x x x x

x x x x

x x x x

x x x

'A A A

'

'A A A

'

'A A A

'

' 'A A A A A

''' '

A A A'

3 ' 'A A A

Tr ( ) ( )

Tr ( ) ( )

Tr ( ) ( )

Tr ( ) Tr ( ) ( )

Tr ( ) ( )

(d x

x x x

x x x

x x x

x x x x

x x x

x 'A

'

A'

A A

) ( )

( )

( )N

x x

x

x

(163)

similarly,

B B B( ) ( )n Nx x (164)

The Hamiltonian of this system is

, ' ', , '

1( )

2H V x x (165)

where V is the molecular interaction between two molecules either of type A or type B. The

Hamiltonian can be re-expressed in the term of density operator:

3 3, ' '

, '

1' ( ) ( ') ( ')

2H d xd x n V nx x x x (166)

This can be proved by substituting eq. 161 into eq. 166 as follows:

3 3, ' '

, '

3 3, ' '

, '

3 3' , '

, '

3 3' , '

, '

3' , '

1' ( ) ( ') ( ')

2

1' ( ) ( ') ( ')

2

1' ( ') ( ') ( )

2

1' ( ') ( ') ( )

2

1' ( ') (

2

H d xd x n V n

d xd x V n

d x n d xV

d x n d xV

d x n V

x x x x

x x x x x

x x x x x

x x x x x

x x x, '

3' , '

, '

3' , '

, '

3, ' '

, '

, ' ', '

, ' ', ' ,

')

1' ( ') ( ')

2

1' ( ' ) ( ')

2

1' ( ') ( ' )

2

1( )

2

1( )

2

d x n V

d x V

d xV

V

V

x x x

x x x x

x x x x

x x

x x (167)

Note that in the above derivation, indexes and are smart enough that they can run over from 0

to NA-1 when =A, and from 0 to NB-1 when =B. Using the linear property of summation and

integral operator in eq. 166, the ensemble average of H respect to is easily obtained as

3 3, ' '

, '

3 3, ' '

, '

1' ( ) ( ') ( ')

2

1' ( ) ( ') ( ')

2

H d xd x n V n

d xd x n V n

x x x x

x x x x

(168)

The ensemble average of ln is calculated as

A A B B A A B B

A A B B A A B

A A B B A A

A A B B B

ln Tr ln

Tr ( ) ( )ln ( ) ( )

Tr ( ) ( ) ln ( ) ln ( )

Tr ( ) ( ) ln ( )

Tr ( ) ( ) ln (

B

x x x x

x x x x

x x x

x x

A A A A B B

)

Tr ( )ln ( ) Tr ( )ln ( )

B

B B

x

x x x x

(169)

The first term in the last line can be further simplified as

A A A A

3A A A A A

3 3A A A A A

3 3A A A A A

3 3A A A A A

3A A

Tr ( )ln ( )

( )ln ( )

( ) ( ) ln ( )

( ) ln ( ) ( )

( ) ln ( ) ( )

( )ln ( )

d x

d x d x

d x d x

d x d x

d x

x x

x x

x x x x

x x x x

x x x x

x x

(170)

Similarly, the second term in the last line of eq. 169 is

3B B B B B BTr ( )ln ( ) ( )ln ( )d xx x x x (171)

Substituting eq. 163, 164, 170 and 171 into 169, we have

3 3A A B B

3 3A A A B B B

A A B B3 3A B

A A B B

A B3A B

A B

ln ( )ln ( ) ( )ln ( )

( )ln ( ) ( )ln ( )

( ) ( ) ( ) ( )ln ln

( ) ( )( ) ln ( )

d x d x

N d x N d x

n n n nd xN d xN

N N N Nn n

d x n nN N

x x x x

x x x x

x x x x

x xx x

(172)

Therefore the variational free energy is

3 3, ' '

, '

A B3A B

A B

1' ( ) ( ') ( ')

2

( ) ( )( ) ln ( )B

F d xd x n V n

n nk T d x n n

N N

x x x x

x xx x

(173)

A similar example about derivation of Debye-Hückel theory for plasma is demonstrated in

the book: Chaikin and Lubensky, Principles of Condensed Matter Physics, 1995, p198-201,

p.204-205.

12. Perturbation theory

Here we take the simplest Landau free energy as an example. The local free energy density is

2 41

2f r u (174)

and the gradient energy is

2

2

c (175)

Thus the Hamiltonian is

23 2 41

2 2

cH d x r u (176)

The partition function is

HZ D e (177)

To calculate Z, we first calculate Z0 which can be explicitly integrated:

00

HZ D e (178)

with H0 being

23 2

01

2 2

cH d x r (179)

Then rewrite eq. 177 to be

0

0

0

0

0

0

0

H V

HV

H

V

ZZ D e

Z

D e eZ

D e

Z e

(180)

If V<<1, we have

2 21

12

1

Ve V V

V (181)

Here V is

3 4V d x u (182)

and

V (183)

The generating function:

30

(0)

( ) ( )

1

20

H d xJ x x

JG J

Z J D e

Z e

(184)

The cumulant expansion. The free energy in SCFT is

30 3

A B, lngc

RF w d r N w V Q w

N (185)

3 ( )

01

( )d xH s

cQ w D P s eV

R

R R (186)

Perturbation expansion:

(0)w w w (187)

1 21 2 31 2 3

(0) (1) (2)

( )(0) 3 31 1 2, ,

1 , ,

1( 1) ( , , , )

n

c c c c

nnc n n

n

Q Q Q Q

Q d r d r C w w wV

r r r (188)

or

1 21 2 31 2 3

(0)

( )3 31 1 2, ,

1 , ,

ln ln

( 1)( , , , )

! n

c c

nn

n nn

V Q V Q

d r d r C w w wn

r r r (189)

13. Weak segregation theory and strong segregation theory

The density field and its conjugated potential field is divided into two parts, the average and

the deviation:

(0) (190)

(0)w w w (191)

Weak segregation theory (WST). The average value of and w are chosen to be

(0) f (192)

(0) 0w (193)

In this case,

, |Q sr r (194)

can be integrated to obtain F(0)

:

(0) (1 )F f f N (195)

The free energy is

(0) (1) (2)3

0

3A B

3 3,

,

(1 )

1

1' , ' ( ) ( ')

2

g

NF F F F

R V

f f N

d r N wV

d rd r C w wV

r r r r

(196)

where F(1)

=0 is applied to the second line. In above expression, only first two order terms are

shown. The WST should expand F to the 4th

-order:

(0) (1) (2) (3) (4)3

0 g

NF F F F F F

R V (197)

Using variation to eliminate w in eq. 196:

0F

w (198)

which leads to a set of equations

0C w C w (199)

Solving above equation to obtain w:

1

wC

(200)

Retaining only the first term of eq.200 leads to RPA, while retaining more terms leads to

Leibler theory.

Strong segregation theory. In strong segregation limit, the interface between two phases is

clear. The free energy can be divided into two parts: bulk and interface. The interfacial free energy

is proportional to the interfacial area A:

in0 6B

FA

k T (201)

The interfacial area is inverse proportional to the phase size:

1

Ad

(202)

SST has a clear physical picture. However, it cannot explain:

(1) N→∞

(2) The joint can move around at the interface, which introduce extra entropy.

14. Saddle point approximation (For more details, see Fredrickson, The Equilibrium Theory

of Inhomogeneous Polymers, 2006, p203-205; Chaikin and Lubensky, Principles of

Condensed Matter Physics, 1995, p198-201, p200-201.)

The partition function can be approximated by the maximum term in the integral

*

H

H

Z D e

e (203)

where * is determined by solving the following variation equation

*| 0H

(204)

It seems that the saddle point approximation can be viewed as one of approaches to

determine variational minima to eq. 153 in section 11. First, we choose a parametrization of in

terms of the order parameter <> of a phase transition. This parametrization must satisfy the

constraints:

Tr 1 (205)

and

Tr (206)

The variational parameter is simply the order parameter <>. If there is no external field in H

coupling linearly to , then F is simply the Helmholtz free energy F(<>). Then eq. 156 turns to

be the form of eq. 191.