Chapter 26 - RADIUS OF GYRATION CALCULATIONS

26:1. SIMPLE SHAPES

26:2. CIRCULAR ROD AND RECTANGULAR BEAM

26:5. GAUSSIAN POLYMER COIL

26:6. THE EXCLUDED VOLUME PARAMETER APPROACH

26:1. SIMPLE SHAPES

4

R

drdr

)d(cosdrr

drdr

rdrd)(cosr

R2

R

0

2π

0

R

0

2π

0

23

2π

0

R

0

2π

0

2R

0

2

2gx

1

1

1

1

2

0

R

0

2

0

R

0

22

2g R

5

3

drrd)sin(

r drrd)sin(

R

)RR(

)RR(

5

3drr4

)RR(4

3R

31

32

51

52

R

R

43

13

2

2g

2

1

Thin disk of radius R:

Sphere of radius R:

Spherical shell:

2

W/2

W/2

W/2

W/2

2

2gx 2

W

3

1

dx

dx x

R

Rectangular plate of width W:

x

y

z

R r

x

y

W

H

x

y

r cos()

rR

cylindrical coordinates

spherical coordinates

cartesiancoordinates

26:2. CIRCULAR ROD AND RECTANGULAR BEAM

Rod of radius R and length L:2222

2g 12

L

2

R

2

L

3

1

2

RR

Rectangular beam of width W, height H and length L :

2222

g 2

L

2

H

2

W

3

1R

Circular Rod

Rectangular Beam

26:5. GAUSSIAN POLYMER COIL

center of mass

i

j

n

i

2i

2g S

n

1R

ijij SSS

n

j,iji

n

j

2j

n

i

2i

n

j,i

2ij S.S2SnSnS

n

j,i

2ij2

n

j,i

2ij2

2g r

n2

1S

n2

1R

|ji|aS 22ij

6

na

n

)1n(

6

ak)

n

k(1

n

a|ji|

n2

aR

222n

k

2n

ji,2

22

g

naR 22n1

Radius of gyration:

End-to-end distance:

Inter-monomer distance:

Note that:

So that:

Radius of gyration:

a: is the segment length

n: is the degree of polymerization

Sijri

Si

26:6. THE EXCLUDED VOLUME PARAMETER APPROACH

222ij |ji|aS

n

j,i

22

2n

j,i

2ij2

2g |ji|

n2

aS

n2

1R

2

22n

k

2

n2)1)(2ν(2ν

ak )

n

k(1

n

a

Self-avoiding walk: = 3/5 5622

g na176

25R

Pure random walk: = 1/2 na2

1R 22

g

Self-attracting walk: = 1/3 3222g na

40

9R

For chains with excluded volume:

Radius of gyration:

12

L

12

nan

2)1)(2ν(2ν

aR

2222

22

g

Thin rigid rod of length L: = 1

COMMENTS

-- Linear plots and empirical models yield radii of gyration.

-- Modeling the radius of gyration is important for data analysis.