The Width of a Complex Ideal Chain

Yanwei Wang, Ole Hassager

Danish Polymer CenterDTU Chemical EngineeringTechnical University of Denmark

External advisers

Financial support

Danish Research Council for Technology and Production Sciences (FTP)

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 details3. Some examples

4. Conclusions

Ideal Chain Statistics

22( , ) ( , )

6P n b P n

n∂

= ∇∂r r

( , )nr

Complex Architecture

linear star pom-pom (two-branch point)

comb ring 8-shaped theta-shaped

tadpole Double-headed tadpole

Double-tailed tadpole

manacles

Double-headed tadpole

Double-tailed tadpole

Branched

Ringed

What is the width of it?

What is the width of it?

u

ir

ˆˆmax( ) min( )i iiiX r u r u= ⋅ − ⋅

ˆˆmax( ) min( )i iiiX r u r u= ⋅ − ⋅

u

ir

The Mean Span Dimension

WHY BOTHER?

Radius of gyration

Hydrodynamic radiusHydrodynamic volume

Size Exclusion Chromatography of Polymers

2-branch point

3-arm star

Linear PE

comb

Sun et al. Macromolecules 37, 4304 (2004)

Wang et al. Macromolecules 43, 1651 (2010)

HOW TO CALCULATE IT?

for an ideal chain but of complex architecture

The basic principles

• Isotropy of a polymer chain in free space

• Identity between one half of the mean span dimension and the depletion layer thickness near a hard wall

• Multiplication rule for independent events.

ˆˆmax( ) min( )i iiiX r u r u= ⋅ − ⋅

u

ir

Wang et al. JCP, 129, 074904 (2008)

ISOTROPY

ˆˆmax( ) min( )

max( ) min( )

i iii

i iii

X r u r u

x x

= ⋅ − ⋅

= −

xmax( )iixmin( )ii

x

max( ) min( )

max( ) min( )

max( ) min( )

i iii

i o o iii

i o o iii

X x x

x x x x

x x x x

= −

= − + −

= − + −

xmax( )iixmin( )ii

x

o

ox

max( ) min( )

2 min( )

i o o iii

o ii

X x x x x

x x

= − + −

= −

xmax( )iixmin( )ii

x

o

ox

max( ) min( )

2 min( )

i o o iii

o ii

X x x x x

x x

= − + −

= −

oα

xmax( )iixmin( )ii

x

o

oxoα

DEPLETIONNEAR A HARD WALL

x

o

ox0x =

x

o

ox0x =

x

o

ox0x =

x

o

ox0x =

( )( )o o oP x H x α= −

x

o

ox0x =

( )( )o o oP x H x α= −

0[1 ( )]o o oP x dx α

∞− =∫

HOW TO CALCULATE ?( )oP xfor an ideal chain but of complex architecture

Three types of fundamental subchains

Arm 1

Arm 2

Loop

Connector

• Arm

• Connector

• Loop

Multiplication rule( ) ( ) ( )

if events A and B are independent P A B P A P B=

o

Arm 1

Arm 2

Loop

Connector

0x = xox

Three base functions

( )Arm ( ; , ) erfP x n b px=2

3x [0, ), x' [0, ), 2

pnb

∈ ∞ ∈ ∞ =

Arm 1

Arm 2

Loop

Connector

( )2 2Loop ( ; , ) 1 exp 4P x n b p x= − −

{ }2 2 2 2Connector 1/ 2( , '; , ) ' exp[ ( ') ] exp[ ( ') ] 'pP x x n b dx p x x p x x dx

π= − − − − +

x0x =

Arm 1

Arm 2

Loop

Connector

ox px

Arm 1 Arm 2 Connector Loop0( ) ( ; , ) ( ; , ) ( , ; , ) ( ; , )o o a o a o p c p l pP x P x n b P x n b P x x n b P x n b d x

∞= ∫

0

1 [1 ( )]2 o oX P x dx

∞= −∫

Wang et al. (2010) submitted

Examples

A linear chain

( )Arm( ) ( ; , ) erfo o oP x P x n b px= =

2

1/ 20

2 82 [1 ( )]3o oNbX P x dx

pπ π∞

= − = =∫

1/ 2

2 1.128382 g

XR π

= ≈16 1.69765

2 3H

XR π

= ≈

A ring

( )2 2Loop( ) ( ; , ) 1 exp 4o oP x P x n b p x= = − −

1/ 2 2

02 [1 ( )]

2 6o oNbX P x dx

pπ π∞

= − = =∫

1.253312 2g

XR

π= ≈ 1.5708

2 2H

XR

π= ≈

A 3-arm star

[ ] ( ) 33Arm( ) ( ; , ) erfo o oP x P x n b px= =

1/ 23/ 20

12 22 [1 ( )] arctan(2 )o oX P x dxpπ

∞ −= − =∫

1.228002 g

XR

≈ 1.720032 H

XR

≈

An f-arm (symmetric) star

[ ] ( )Arm

0

( ) ( ; , ) erf

2 [1 ( )]

ffo o o

o o

P x P x n b px

X P x dx∞

= =

= −∫

An f-arm (symmetric) star

[ ] ( )Arm

0

( ) ( ; , ) erf

2 [1 ( )]

ffo o o

o o

P x P x n b px

X P x dx∞

= =

= −∫

0 5 10 15 201.0

1.2

1.4

1.6

1.8

2.02 H

XR

2 g

XR

2-branch point

3-arm star

Linear PE

comb

Sun et al. Macromolecules 37, 4304 (2004)

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.