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

2. Nano- and Microfluidic Devices
• Faster analyses & fewer materials reduces costs; small scale increases convenience
• Applications: Genomics (DNA/protein separation and sequencing), pharmaceutical screening, sensors, clinical analysis, biomedical implants
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)

4. Polyelectrolytes --[--CH---CH2--]n-- | SO3-H+
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) > 103
Contour length > 1 mm

5. Current Work • Layer-by-layer assembly of polyelectrolytes
• Polymer electrophoresis in narrow channels
DNA motion
ts = nm
td = 1.5-3mm
(-)
(+)
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

7. Experimental observation of a complex electro-osmotic flow
k-1 ~ 1nm << 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 patternedx
s+=sosin(qx)
s-=sosin(qx) l 2h
Eext z
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 > 103
Contour length > 1 mm r1 r2 rN
• Polymers perform a random walk in athermal solvents
Kuhn step: characteristic step size of random walk, ~ nm
Coarse-grained model: beads and rods; 1 rod = 1 Kuhn step

12. Langevin equation Viscous drag felt at the beads
Solvent molecules exert a random Brownian force
Bead inertia neglected r1 r2 rN
Bead-rod model
• Force balance on each bead

13. 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

14. Entropic spring • Replace bead-rod chain by an effective springrN
• Fewer beads, but can’t 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

15. Bead-spring model Viscous drag felt at the beads Kuhn steps
Each spring represents many
Kuhn steps b2 b1 b3 bN
• Force balance on each bead

16. Terms in the force balance
li = Ri/NK,sbK
Ri = |bi+1 - bi|

17. 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 = mm, z/kBT = 1 s/mm2
20 beads/chain, t = 0.7 s, Eext = 9.5 kV/m
Contour length ~ 67 mm

18. 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 patternedx
s+=sosin(qx)
s-=sosin(qx) l 2h
Eext z
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
Maximum possible value is 36100; equilibrium value is 19

21. Heterogeneity in end-to-end distance among ensemble members
Time
Values range from 150 to 25000

22. Histogram of trajectories
Number of trajectories
Percentage extension

23. End-to-end distance and local Weissenberg number
Wix,x
Time

24. Spatial position of trajectories
Distance moved along z-direction
t = 90
t = 180
t = 150
t = 120
Distance moved along x-direction

25. Streamlines when both walls are patternedx
s+=sosin(qx)
s-=sosin(qx) l 2h
Eext z
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

27. 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

28. Distance moved along the x-direction
Charged polymer
Distance moved along
the z-direction
Distance moved along the x-direction
<l2>
Time
Distance moved along the x-direction
Time

29. 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

30. Current Work • Layer-by-layer assembly of polyelectrolytes
• Polymer electrophoresis in narrow channels
DNA motion
ts = nm
td = 1.5-3mm
(-)
(+)
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