Polymer Dynamics in Microfluidic Systems

1
Brownian Dynamics Simulations of Polymer
Behavior in Nano- and Microfluidic Systems
Satish Kumar Department of Chemical Engineering
and Materials Science University of Minnesota

2
Nano- and Microfluidic Devices
Lab-Chip
Burns et. al., Science 282 (1998) 484
Caliper Tech. and Agilent
3
Nature of Flows in Microchannels

• Smallest channel dimension is 1 mm or less
• Viscous forces dominate over inertial forces
  (low Reynolds number)
• Surface tension forces often significant
  
• Flows driven by electric fields or pressure
  gradients
• Polymer solutions often handled (e.g., DNA,
  proteins)

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Polyelectrolytes
Polymers whose monomers contain functional groups
that become ionized when placed in an aqueous
solution
----CH---CH2--n-- SO3-H
Poly(vinylsulfonic acid)
Monomer length 1 nm Number of monomers (n) gt
103 Contour length gt 1 mm
5
Current Work

• Layer-by-layer assembly of polyelectrolytes

Polymer electrophoresis in narrow channels
Han and Craighead (2000)
6
Brownian Dynamics Simulations of Polymer
Stretching and Transport in a Complex
Electro-osmotic Flow
Ajay S. Panwar and Satish Kumar Department of
Chemical Engineering and Materials
Science University of Minnesota
J. Chem. Phys. 118 (2003) 925
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Experimental observation of a complex
electro-osmotic flow
k-1 1nm ltlt a, b (200mm), l (400mm) Velocities
100 mm/s strain rates 1 s-1
Stroock et. al., Phys. Rev. Lett. 84 (2000) 3314
8
Streamlines when both walls are patterned
Stagnation point
Ajdari, Phys. Rev. Lett. 75 (1995) 755 Ajdari,
Phys. Rev. E 53 (1996) 4996
9
Goals of this work

• Determine the effectiveness of the stagnation
  point in stretching polymers
• Characterize polymer dynamics in a model flow
  with an inhomogeneous velocity gradient
• Examine the competition between electroosmosis
  and electrophoresis on polymer transport

10
Governing equations for velocity field
Stokes equation
Incompressibility
No-slip/penetration BC
Debye-Hückel equation
BC on potential
11
Mechanical models of polymers
Typical scales Monomer length 1 nm Number of
monomers gt 103 Contour length gt 1 mm
Polymers perform a random walk in athermal
solvents Kuhn step characteristic step size of
random walk, 1 - 100 nm Coarse-grained model
beads and rods 1 rod 1 Kuhn step
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Langevin equation

• Viscous drag felt at the beads
• Solvent molecules exert a random Brownian force
• Bead inertia neglected

Bead-rod model
Force balance on each bead
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Can we coarse-grain even more?
How much force is required to separate the ends
of a bead-rod chain by a certain distance?
Neglect internal energy S k ln W A
-TS dA F dR R rN-r1
Force
Can replace bead-rod chain by a spring
Relative extension
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Entropic spring
Replace bead-rod chain by an effective spring
Fewer beads, but cant capture changes in
conformation and orientation along polymer
backbone
Compromise replace bead-rod chain by a series
of springs, where each spring represents many
Kuhn steps
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Bead-spring model

• Viscous drag felt at the beads
• Each spring represents many
• Kuhn steps

b2
b1
b3
bN
Force balance on each bead
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Terms in the force balance
li Ri/NK,sbK
Ri bi1 - bi
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Scaling and parameter values
Length
b (NK,sbK2)1/2
Time
zb2/kBT
Velocity
m0Eext m0 s0/hk
Force
kBT/b
NK,s 100, bK 0.033 mm, z/kBT 1 s/mm2 20
beads/chain, t 0.7 s, Eext 9.5 kV/m Contour
length 67 mm
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Assumptions
Reflecting boundary conditions at the walls
No hydrodynamic interactions Neglect
intramolecular interactions Low zeta
potential at wall Boltzmann distribution for
counterions
19
Streamlines when both walls are patterned
Stagnation point
Ajdari, Phys. Rev. Lett. 75 (1995) 755 Ajdari,
Phys. Rev. E 53 (1996) 4996
20
Ensemble-averaged mean square end-to-end
distance vs. time
Time
Maximum possible value is 36100 equilibrium
value is 19
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Heterogeneity in end-to-end distance among
ensemble members
Time
Values range from 150 to 25000
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Histogram of trajectories
Number of trajectories
Percentage extension
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End-to-end distance and local Weissenberg number
l2
Wix,x
Time
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Spatial position of trajectories
t 40
Distance moved along z-direction
t 90
t 180
t 150
t 120
Distance moved along x-direction
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Streamlines when both walls are patterned
Stagnation point
Ajdari, Phys. Rev. Lett. 75 (1995) 755 Ajdari,
Phys. Rev. E 53 (1996) 4996
26
Trapping of trajectories
200 time units 85 of trajectories remain in
first recirculation region 1000 time units
80 of trajectories remain in first recirculation
region Flows are possibly useful for
localizing positions of macromolecules
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Influence of Brownian history
Why are the trajectories different different
initial conditions or different Brownian
histories? Different initial conditions same
Brownian history Different conformations at
early times Similar conformations at later
times Same initial conditions different
Brownian histories Wide distribution of
conformations Brownian history primarily
controls stretching in these flows
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Charged polymer
Distance moved along the z-direction
Distance moved along the x-direction
ltl2gt
Distance moved along the x-direction
Time
Time
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Conclusions

• These complex electroosmotic flows are not as
  effective at stretching polymers as pure
  extensional flows
• Polymers get convected to wall amount of
  stretching is proportional to amount of time
  spent there
• Stretching in these flows primarily controlled by
  Brownian history of polymer, not initial
  conformation
• Charged polymers become trapped in recirculation
  regions below a critical charge density
• Flows may be useful for localizing position of
  polymers in microfluidic devices

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Current Work

• Layer-by-layer assembly of polyelectrolytes

Polymer electrophoresis in narrow channels
Han and Craighead (2000)
31
Acknowledgments

• American Chemical Society Petroleum Research Fund
• Industrial Partnership for Research in
  Interfacial and Materials Engineering (IPRIME)
• Army High Performance Computing Research Center
• 3M Nontenured Faculty Award
• Shell Faculty Career Initiation Award