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

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

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)

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

Current Work • Layer-by-layer assembly of polyelectrolytes • Polymer electrophoresis in narrow channels DNA motion ts = 75-100nm td = 1.5-3mm (-) (+) Han and Craighead (2000)

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

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

Streamlines when both walls are patterned x 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

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

Governing equations for velocity field Stokes’ equation Incompressibility No-slip/penetration BC Debye-Hückel equation BC on potential

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, ~1 - 100 nm Coarse-grained model: beads and rods; 1 rod = 1 Kuhn step

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

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

Entropic spring • Replace bead-rod chain by an effective spring rN • 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

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

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

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

Assumptions • Reflecting boundary conditions at the walls • No hydrodynamic interactions • Neglect intramolecular interactions • Low zeta potential at wall • Boltzmann distribution for counterions

Streamlines when both walls are patterned x 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

Ensemble-averaged mean square end-to-end distance vs. time Maximum possible value is 36100; equilibrium value is 19

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

Histogram of trajectories Number of trajectories Percentage extension

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

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

Streamlines when both walls are patterned x 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

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

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

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

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

Current Work • Layer-by-layer assembly of polyelectrolytes • Polymer electrophoresis in narrow channels DNA motion ts = 75-100nm td = 1.5-3mm (-) (+) Han and Craighead (2000)

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