A general method for calculating the mean span dimensions of various branched and ringed polymers under the assumption of Gaussian chain statistics.

1. The Width of a Complex Ideal Chain
Yanwei Wang, Ole Hassager
Danish Polymer Center
DTU Chemical Engineering
Technical University of Denmark

2. External advisers
Financial support
Danish Research Council for Technology and Production Sciences (FTP)

3. Outline
1. Introduction
–What is a complex ideal chain?
–What is the width of it?
–Why bother?
2. Method
–Principle
–Base functions
–The rest are details
3. Some examples
4. Conclusions

4. (r, n)
∂P (r, n) b 2
2
= ∇ P (r, n)
∂n 6
Ideal Chain Statistics

5. linear star pom-pom
(two-branch point)
Ringed
comb ring 8-shaped theta-shaped
Branched
tadpole Double-headed Double-tailed
tadpole tadpole
manacles
Complex Architecture

6. What is the width of it?

7. What is the width of it?

8. 
ri
ˆ
u
 
=
X max(ri ⋅ u ) − min(ri ⋅ u )
ˆˆ
i i

9. 
ri
ˆ
u
 
=
X max(ri ⋅ u ) − min(ri ⋅ u )
ˆˆ
i i
The Mean Span Dimension

10. WHY BOTHER?

11. Hydrodynamic volume Hydrodynamic radius
Radius of gyration

12. Size Exclusion
Chromatography
of Polymers

13. Linear PE
3-arm star
2-branch point
comb
Sun et al. Macromolecules 37, 4304 (2004)

14. Wang et al. Macromolecules 43, 1651 (2010)

15. HOW TO CALCULATE IT?
for an ideal chain but of complex architecture

16. The basic principles
• Isotropy of a polymer chain in
free space
 • Identity between one half of
ri
the mean span dimension
and the depletion layer
thickness near a hard wall
ˆ
u Wang et al. JCP, 129, 074904 (2008)
 
=
X max( ri ⋅ u ) − min(ri ⋅ u )
ˆˆ
i i
• Multiplication rule for
independent events.

17. ISOTROPY

18.  
=
X max(ri ⋅ u ) − min(ri ⋅ u )
ˆˆ
i i
max( xi ) − min( xi )
i i
min( xi ) max( xi ) x
i i

19. X max( xi ) − min( xi )
i i
= max( xi ) − xo + xo − min( xi )
i i
= max( xi ) − xo + xo − min( xi )
i i
o
min( xi ) xo max( xi ) x
i i

20. =
X max( xi ) − xo + xo − min( xi )
i i
= 2 xo − min( xi )
i
o
min( xi ) xo max( xi ) x
i i

21. =
X max( xi ) − xo + xo − min( xi )
i i
= 2 xo − min( xi )
i
αo
o
αo
min( xi ) xo max( xi ) x
i i

22. DEPLETION
NEAR A HARD WALL

23. o
x=0 xo x

24. o
x=0 xo x

25. o
x=0 xo x

26. o
=
P( xo ) H ( xo − α o )
x=0 xo x

27. o
=
P( xo ) H ( xo − α o )
∞
∫
0
αo
[1 − P( xo )]dxo =
x=0 xo x

28. HOW TO CALCULATE P( xo ) ?
for an ideal chain but of complex architecture

29. Three types of
fundamental subchains
Arm 1
• Arm
Loop
• Connector Arm 2
Connector
• Loop

30. Arm 1
Loop
o
Arm 2
Connector
x=0 xo x
Multiplication rule
P( A  B) = P( A) P( B)
if events A and B are independent

31. Three base functions
Arm 1
Loop
Arm 2
Connector
PArm ( x; n, b) = erf ( px ) 3
x ∈ [0,∞), x' ∈ [0,∞), p = 2
2nb
PLoop ( x; n, b) =exp ( −4 p 2 x 2 )
1−
PConnector ( x, x '; n, b)=
dx '
p
π 1/ 2
{exp[− p ( x − x ') ] − exp[− p ( x + x ') ]} dx '
2 2 2 2

32. Arm 1
Loop
Arm 2
Connector
x=0 xo xp x
∞
P( xo ) = PArm ( xo ; na1 , b) PArm ( xo ; na 2 , b) ∫ PConnector ( xo , x p ; nc , b)PLoop ( x p ; nl , b)d px
0
1 ∞
2
=X ∫ 0
[1 − P ( xo )]dxo
Wang et al. (2010) submitted

33. Examples

34. A linear chain
= PArm ( xo ; n, b) erf ( pxo )
P( xo ) =
∞ 2 8 Nb 2
2∫
X = [1 − P ( xo )]dxo = =
0 π
1/ 2
p 3π
X 2 X 16
= 1/ 2 ≈ 1.12838 = ≈ 1.69765
2 Rg π 2 RH 3π

35. A ring
P( xo ) = ( xo ; n, b) =exp ( −4 p 2 x 2 )
PLoop 1−
∞ π 1/ 2 π Nb 2
2 ∫ [1
X = − P ( xo )]dxo ==
0 2p 6
X π X π
= ≈ 1.25331 = ≈ 1.5708
2 Rg 2 2 RH 2

36. A 3-arm star
[=
PArm ( xo ; n, b)] erf ( pxo ) 
3 3
P( xo )  
∞ 12 2
2∫
X = [1 − P ( xo )]dxo =2 arctan(2−1/ 2 )
0 π 3/ p
X X
≈ 1.22800 ≈ 1.72003
2 Rg 2 RH

37. An f-arm (symmetric) star
= [ = erf ( pxo ) 
P( xo ) PArm ( xo ; n, b)] 
f f

∞
= 2∫ [1 − P( xo )]dxo
X
0

38. An f-arm (symmetric) star
= [ = erf ( pxo ) 
P( xo ) PArm ( xo ; n, b)] 
f f
2.0 X 
2R
1.8 = H2 ∞ [1 − P( x )]dx
X ∫ o o
0
1.6
X
1.4 2 Rg
1.2
1.0
0 5 10 15 20

39. Linear PE
3-arm star
2-branch point
comb
Sun et al. Macromolecules 37, 4304 (2004)

40. Conclusions
• A general method is developed for calculating the
width (mean span dimension) of polymer chains
assuming ideal chain statistics.
• The method comes from
– Isotropy of a polymer chain in free space
– Polymer depletion near a hard wall
– Multiplication rule for independent events.
• The method can be routinely applied to any
complicated chain architectures.