Chapter 28 – FORM FACTORS FOR POLYMER SYSTEMS

28:1. THE DEBYE FUNCTION FOR GAUSSIAN CHAINS

28.2. SINGLE-CHAIN FORM FACTOR FOR GAUSSIAN CHAINS

28.3. OTHER POLYMER CHAIN ARCHITECTURES

Gaussian probability distribution:

2ij

2ij

3/2

2ij

ijr2

3rexp

r 2π

3)r(P

28:1. THE DEBYE FUNCTION FOR GAUSSIAN CHAINS

|ji|ar 22ij Inter-monomer mean square distance:

n

ji,ij2]r.Qiexp[

n

1)Q(P

n

j,i

2ij

2

2 6

rQexp

n

1

n

j,i

22

2 6

|ji|aQexp

n

1

n

ji,

n

1k

)k(F)kn(2n|)ji(|F

n

1k

22

2 6

kaQexp)kn(2n

n

1)Q(P

2g

22g

24

g4

RQ1)RQexp(RQ

2)Q(P

6/naR 2g

Form factor for Gaussian chains:

Use the identity:

Form factor:

The Debye function:

Radius of gyration:

28.2. SINGLE-CHAIN FORM FACTOR FOR GAUSSIAN CHAINS

2ij

2ij

3/2

2ij

ijr2

3rexp

r 2π

3)r(P

222ij |ji|ar

n

j,i

2ν22

2|ji|

6

aQexp

n

1)Q(P

2ν22n

1k2

k6

aQexpk)-(n2

n

1)Q(P

2ν2

221

0

xn6

aQexp)x1( dx2)Q(P

1dU

0

t)texp(dt)U,d(

U),ν

1γ(

νU

1 - U),

2ν

1γ(

νU

1)Q(P

1/ν1/2ν

Gaussian probability distribution:

Gaussian chains with excluded volume:

Form factor with excluded volume:

Form factor:

Define the Incomplete Gamma function:

Final result:

Continuous limit:

/1Q

1)Q(P

28.3. OTHER POLYMER CHAIN ARCHITECTURES

2g

22g

24

g4

RQ1)RQexp(RQ

2)Q(P

n

i

22

6

|1i|aQexp

n

1)Q(F

2

g2

2g

2

RQ

RQexp1

]RQexp[)Q(E 2g

2

Form factor:

Form factor amplitude:

Propagation factor:

)Q(F)Q(E)Q(F)Q(P CBAAC

P(Q)

F(Q)

E(Q)

ij

1

j

1

N

Case of a triblock copolymer:

AB

C

Scattering cross section (cm-1): )Q(PV)Q(PVV

N

d

)Q(dP

22P

2

Where: (N/V) is the particle (or polymer) number density is the particle volume fractionVP is the particle (or polymer) volume2 is the contrast factorP(Q) is the form factorSI(Q) is the structure factor

SCATTERING CROSS SECTION

COMMENTS

-- Form factors for polymers are mass fractals.

-- Their calculation is needed for modeling of polymeric systems of various architectures.