Separation of Macromolecules by Their Size: The Mean Span Dimension
Transcript of Separation of Macromolecules by Their Size: The Mean Span Dimension
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]
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Collaborations
Financial support
Danish Research Council for Technology and Production Sciences (FTP)
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Size Exclusion Chromatography (SEC)
~ 50 cm
~ 1 cm
http://en.wikipedia.org/wiki/Size-exclusion_chromatography
Polymer Laboratories’ Technical Bulletin
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Micro- and Nano-Fluidics
Baba et al. (2003) Sano et al. (2003)
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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
!
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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!=
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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