Separation of Macromolecules by Their Size:The Mean Span Dimension

Yanwei Wang, Ole Hassager*

Danish Polymer CenterDepartment of Chemical and Biochemical EngineeringTechnical University of Denmark, DK-2800, Lyngby, Denmark*EMAIL: oh@kt.dtu.dk

10/24/102

Collaborations

Financial support

Danish Research Council for Technology and Production Sciences (FTP)

10/24/103

Size Exclusion Chromatography (SEC)

~ 50 cm

~ 1 cm

http://en.wikipedia.org/wiki/Size-exclusion_chromatography

Polymer Laboratories’ Technical Bulletin

10/24/104

Micro- and Nano-Fluidics

Baba et al. (2003) Sano et al. (2003)

10/24/105

Classical Transport Model

At the slower rate the resolution was improved, butthe peaks were not shifted.

Moore (1964) Experiments:

Polystyrene in toluene

s108: Mw=267 kg/mol, Rg ≈ 30 nm

s102: Mw=82 kg/mol, Rg ≈ 15 nm

Equilibrium + Dispersion in the fluid phase

V0

!c

!t+V

p

!q

!t= D

TV0

!2c

!x2" uV0

!c

!x

Equilibrium partition coefficient:

K = q / c

Ve= V0 + K V

p

Number of stages N given by:

N

2= Pé =

uL

DT

~D

uL

cqVp

V0 u

stationary

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Cavity Péclect Number

U

W

2/

/

g

c

R DPe

W U=

0c

Pe =

10c

Pe !

Hongbo Ma (2007), Ph.D Thesis

Hernandez-Ortiz et al. (2008)

Rg! 20 nm, W ! 100 nm, D ! 10!11!11 m2 /s

Pec"1 requires that U " 9 m/hour

For a column of cross section area ~ 1 cm2,

the critical flow rate is ~ 900 ml/hour, which

is much higher than typical flow rates used.

But for a typical SEC experiment:

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Equilibrium Partition Coefficient

K =

cencentration in a pore

bulk concentration= exp !S / k

B"# $%

Dilute polymer solutions

Enthalpic interactions are negligible. Thus, steric exclusions only

Shaded molecules: allowed configurations // Open molecules: forbidden configurations

Loss of entropy due to confinement:

lnB

A T S k T K! = " ! = " Free energy of confinement:

Giddings et al. (1968), Casassa (1967, 1969, 1976)

Reviews: Gorbunov & Skvortsov (1995), Teraoka (1996)

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Scaling Laws

Exclusion limit

Blob theory

!A ! R

gd( )

1/"

Weak Confinement

!A ! R

gd

K ! 0.95 Total Permeation

K ! 0.05

Wang et al. (2010)

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Confinement Regimes

Graham (2011)

Vp

: pore volume

Sp

: inner surface area

Depletion layer thickness:

! ! Rg

Confinement length scale

d !Vp

/ Sp>> !

Equilibrium partition coefficient:

K =

Vp" S

p!

Vp

= 1"S

p

Vp

!

Confinement free energy:

#A = "kBT ln K " ! d " R

gd

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Depletion Layer Thickness

y!

com

The probability of success in generating a chain with its centerof mass located at distance y from an absorbing wall:

( ) H( )y y! "= #

Depletion layer thickness:

0[1 ( )]y dy! " #

$= % =& Wang et al. (2008)

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Mean Span Dimension

!

com

Daniels (1941), Kuhn (1945, 1948)

Wang et al. (2008), Wang et al. (submitted)

!

com

com

min( )

max( )

max( ) min( )

1max( ) min( )

2

ii

ii

i iii

i iii

y y

y y

y y

y y

!

"

! "

# !

= $

= $

+ = $

= = $

The mean span dimension, X

Feret's Statistical Diameter(The mean caliper diameter)

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Universality

1p

p

SK

V!= "

112

p

p

SK X

V= !

1

2X! =

1

2

p

p

SX

V

K1

12

p

p

SK X

V= !

Results fordifferent

modelpolymers

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K

2Rg/d

K

2RH/d

K

/dX

Partition coefficients as a function ofpolymer size to pore size.

Rg: Radius of gyration

RH: Hydrodynamic radius

: Mean span dimension

Slit of width d

X

Linear chains ofdifferentnumber ofsegments

Branched chains

Linear chains ofdifferentnumber ofsegments

Branched chains

Linear chains ofdifferentnumber ofsegments

Branched chains

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K

2Rg/d

K

2RH/d

K

/dX

Square channel ofwidth d

Linear chains of differentnumber of segments

Branched chains

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Cylinder ofdiameter d

Linear chains of differentnumber of segments

Branched chains

K

2Rg/d

K

2RH/d

K

/dX

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K

2Rg/d

K

2RH/d

K

/dX

Cubic box ofwidth d

Linear chains of differentnumber of segments

Branched chains

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K

2Rg/d

K

2RH/d

K

/dX

Spherical cavityof diameter d

Linear chains of differentnumber of segments

Branched chains

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Experiments

Sun et al. (2004)

2-branch point

3-arm star

Linear PE

comb

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Wang et al. (2010)

was estimated from experimental -data and the ratio /

calculated from random walk simulations.

g gX R X R

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Experimental “Universal calibration” Benoit et al. (1967)

Sun et al. (2004)

2-branch point

3-arm star

Linear PE

comb

2[ ]

5hd

A

MV

N!=

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Conclusions & Further Questions

• Modeling Size Exclusion Chromatography– Equilibrium partitioning– Weak confinement regime– Dilute solution– Steric exclusion only– The mean span dimension matters!

• The mean span dimension– Easy to calculate– Challenge 1: How to measure the mean span dimension of a

polymer chain?

• The hydrodynamic volume– Easy to measure experimentally– Challenge 2: How is the mean span dimension connected to the

hydrodynamic volume, if there is a connection?

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Supporting information

17/04/2008Presentation name23 DTU Chemical Engineering,Technical University of Denmark

Calculations of the Mean Span Dimension

• Ideal Gaussian chains • Non-ideal chains

An integral formulation hasbeen developed.

Wang et al. (submitted)

Simulations are necessary.

/(2 )H

X R

/(2 )g

X R

!

17/04/2008Presentation name24 DTU Chemical Engineering,Technical University of Denmark

Branching Parameters

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Hydrodynamic Volume

IUPAC Compendium of Chemical Terminology 2nd Edition (1997)

Hydrodynamic volume: the volume of a hydrodynamicallyequivalent sphere.

Hydrodynamically equivalent sphere: A hypothetical sphere,impenetrable to the surrounding medium, displaying in ahydrodynamic field the same frictional effect as an actual polymermolecule.

The size of a hydrodynamically equivalent sphere may be differentfor different types of motion of the macromolecule, e.g. fordiffusion and for viscous flow.

2[ ]

5hd

A

MV

N!=

17/04/2008Presentation name26 DTU Chemical Engineering,Technical University of Denmark

How to measure the mean span dimensionof a polymer chain

Hsieh & Doyle (2008)Image analysis

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Baba et al. (2003)Sano et al. (2003)

Your chip as a macromolecular caliper?

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The Mean Span Dimension from SEC

10/24/1029Radius of gyration

Hydrodynamic radiusHydrodynamic volume

Could the mean span dimension be the 4th wheel inpolymer characterization