Separation of Macromolecules by Their Size: The Mean Span Dimension Yanwei Wang, Ole Hassager* Danish Polymer Center Department of Chemical and Biochemical Engineering Technical University of Denmark, DK-2800, Lyngby, Denmark *EMAIL: [email protected]
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Separation of Macromolecules by Their Size: The Mean Span Dimension
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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: [email protected]
10/24/102
Collaborations
Financial support
Danish Research Council for Technology and Production Sciences (FTP)
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)
10/24/1011
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)
10/24/1012
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
10/24/1013
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
10/24/1014
K
2Rg/d
K
2RH/d
K
/dX
Square channel ofwidth d
Linear chains of differentnumber of segments
Branched chains
10/24/1015
Cylinder ofdiameter d
Linear chains of differentnumber of segments
Branched chains
K
2Rg/d
K
2RH/d
K
/dX
10/24/1016
K
2Rg/d
K
2RH/d
K
/dX
Cubic box ofwidth d
Linear chains of differentnumber of segments
Branched chains
10/24/1017
K
2Rg/d
K
2RH/d
K
/dX
Spherical cavityof diameter d
Linear chains of differentnumber of segments
Branched chains
10/24/1018
Experiments
Sun et al. (2004)
2-branch point
3-arm star
Linear PE
comb
10/24/1019
Wang et al. (2010)
was estimated from experimental -data and the ratio /
calculated from random walk simulations.
g gX R X R
10/24/1020
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!=
10/24/1021
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?
10/24/1022
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
10/24/1025
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
10/24/1027
Baba et al. (2003)Sano et al. (2003)
Your chip as a macromolecular caliper?
10/24/1028
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