^{1}, Juan J. de Pablo

^{1}and Michael D. Graham

^{1,a)}

### Abstract

This work presents an immersed boundary method that allows fast Brownian dynamics simulation of solutions of polymer chains and other Brownian objects in complex geometries with fluctuating hydrodynamics. The approach is based on the general geometry Ewald-like method, which solves the Stokes equation with distributed regularized point forces in *O*(*N*) or operations, where *N* is the number of point forces in the system. Time-integration is performed using a midpoint algorithm and Chebyshev polynomial approximation proposed by Fixman. This approach is applied to the dynamics of a genomic DNA molecule driven by flow through a nanofluidic slit with an array of nanopits on one wall of the slit. The dynamics of the DNA molecule was studied as a function of the Péclet number and chain length (the base case being λ-DNA). The transport characteristics of the hopping dynamics in this device differ at low and high Péclet number, and for long DNA, relative to the pit size, the dynamics is governed by the segments residing in the pit. By comparing with results that neglect them, hydrodynamic interactions are shown to play an important quantitative role in the hopping dynamics.

The authors acknowledge support from the University of Wisconsin-Madison Nanoscale Science and Engineering Center, which is funded by the National Science Foundation (NSF) (DMR-0425880). The authors thank Amit Kumar, Pratik Pranay, and Juan Hernandez-Ortiz for useful discussions.

I. INTRODUCTION

II. METHODS FOR HYDRODYNAMICS OF CONFINED POLYMER SOLUTIONS

III. POLYMERMODEL AND SIMULATION METHOD

A. Model and governing equations

B. Governing equations

C. Mobility tensor and time-integration algorithm

D. Chebyshev approximation

E. Fast Stokes solver with complex boundary conditions

1. Immersed boundary method

2. Periodic GGEM

3. Local velocity field

4. Global velocity field

5. Validation: GGEM

6. Validation: IBM

IV. DNA FLOWING ACROSS AN ARRAY OF NANOPITS

A. Dynamics at low Péclet number

B. Dynamics at high Péclet number

V. CONCLUSIONS

### Key Topics

- Polymers
- 61.0
- DNA
- 41.0
- Hydrodynamics
- 27.0
- Rheology and fluid dynamics
- 18.0
- Hydrological modeling
- 16.0

## Figures

DNA flowing through a nanoslit with embedded nanopit arrays (reproduced with permission from Ref. 6.) (a) Scanning electron micrograph of typical 1 × 1 μm square pits embedded in a nanoslit at 1 μm intervals. (b) Epifluorescent images of stained λ-DNA molecules confined in a 107 nm deep slit embedded with 1 × 1 μm pits spaced 2 μm apart. (c) Fluorescent images of λ-DNA traveling across a nanopit array under an applied pressure of 40 mbar in the device shown in (b).

DNA flowing through a nanoslit with embedded nanopit arrays (reproduced with permission from Ref. 6.) (a) Scanning electron micrograph of typical 1 × 1 μm square pits embedded in a nanoslit at 1 μm intervals. (b) Epifluorescent images of stained λ-DNA molecules confined in a 107 nm deep slit embedded with 1 × 1 μm pits spaced 2 μm apart. (c) Fluorescent images of λ-DNA traveling across a nanopit array under an applied pressure of 40 mbar in the device shown in (b).

Schematic of the domain and distribution of polymer beads and boundary nodes. The domain is a rectangular parallelepiped of dimensions *L* _{ x } × *L* _{ y } × *L* _{ z } in which the flow satisfies periodic boundary conditions. The beads of a polymer molecule are shown by the filled symbols, and their positions are contained in the vector **R**. The no-slip boundary ∂Ω_{ b } is represented by regularized point forces located at boundary nodes **R** _{ b }, which are separated with a characteristic spacing *h*. The computational grid for solution of the “global” flow problem is regular, with grid spacings determined by the screening parameter α in the GGEM splitting of the force density, Eqs. (24)–(26).

Schematic of the domain and distribution of polymer beads and boundary nodes. The domain is a rectangular parallelepiped of dimensions *L* _{ x } × *L* _{ y } × *L* _{ z } in which the flow satisfies periodic boundary conditions. The beads of a polymer molecule are shown by the filled symbols, and their positions are contained in the vector **R**. The no-slip boundary ∂Ω_{ b } is represented by regularized point forces located at boundary nodes **R** _{ b }, which are separated with a characteristic spacing *h*. The computational grid for solution of the “global” flow problem is regular, with grid spacings determined by the screening parameter α in the GGEM splitting of the force density, Eqs. (24)–(26).

Properties of **M** ^{ bb } for the case where a rigid sphere with radius *R* = 3 is represented by uniformly distributed points on the surface and the parameters for the calculation are *L* _{ x } = *L* _{ y } = *L* _{ z } = 10, Δ*x* = Δ*y* = Δ*z* = 0.25, ξ = 4.0. (a) Condition number as a function of number of points on the sphere shell. (b) Relative residual as a function of number of iterations.

Properties of **M** ^{ bb } for the case where a rigid sphere with radius *R* = 3 is represented by uniformly distributed points on the surface and the parameters for the calculation are *L* _{ x } = *L* _{ y } = *L* _{ z } = 10, Δ*x* = Δ*y* = Δ*z* = 0.25, ξ = 4.0. (a) Condition number as a function of number of points on the sphere shell. (b) Relative residual as a function of number of iterations.

Screening function *g*(α*r*) (Eq. (27)) for the GGEM algorithm.

Screening function *g*(α*r*) (Eq. (27)) for the GGEM algorithm.

Comparison of the *x*-component of the velocity field driven by a point force (lines) acting along **+** ** x **-axis with that generated by a regularized point force (symbols) with modified Gaussian form (Eq. (27), α = 2.0). (Top) velocity along

*x*-axis. (Bottom) velocity along

*y*-axis.

Comparison of the *x*-component of the velocity field driven by a point force (lines) acting along **+** ** x **-axis with that generated by a regularized point force (symbols) with modified Gaussian form (Eq. (27), α = 2.0). (Top) velocity along

*x*-axis. (Bottom) velocity along

*y*-axis.

Comparison of Hasimoto's result^{37} (symbols) and numerical solution (lines) for the *x* component of velocity due to a point force acting along +*x* direction in a periodic domain.

Comparison of Hasimoto's result^{37} (symbols) and numerical solution (lines) for the *x* component of velocity due to a point force acting along +*x* direction in a periodic domain.

Error ||*E*||_{2} as a function of (a) screening parameter α and (b) mesh size Δ*x* for a point force at the center of a cubic periodic domain.

Error ||*E*||_{2} as a function of (a) screening parameter α and (b) mesh size Δ*x* for a point force at the center of a cubic periodic domain.

(a) Velocity profile of laminar flow through a slit. The line represents the analytic profile and the symbols represent the numerical result by IBM. (b) Error of IBM as a function of boundary grid size *h* and the slope is 1.22.

(a) Velocity profile of laminar flow through a slit. The line represents the analytic profile and the symbols represent the numerical result by IBM. (b) Error of IBM as a function of boundary grid size *h* and the slope is 1.22.

Validation of GGEM/IBM algorithm. Open symbols are numerical results and lines are analytical calculations. (a) Velocity profile *u* _{ x } around a unit sphere with no-slip boundary condition. Open circles represent points along x-axis and open squares are for points along *y*-axis. (b) Stokeslet near a single wall. Velocity profile of *u* _{ x } along +*z* direction for various (*x*, *y*) pairs. Open squares are for (*x*, *y*) = (3, 3) and open circles are for (*x*, *y*) = (1, 1).

Validation of GGEM/IBM algorithm. Open symbols are numerical results and lines are analytical calculations. (a) Velocity profile *u* _{ x } around a unit sphere with no-slip boundary condition. Open circles represent points along x-axis and open squares are for points along *y*-axis. (b) Stokeslet near a single wall. Velocity profile of *u* _{ x } along +*z* direction for various (*x*, *y*) pairs. Open squares are for (*x*, *y*) = (3, 3) and open circles are for (*x*, *y*) = (1, 1).

Schematic (top) and discretized representation (bottom) of the nanopit problem. Gray spheres indicate points at which no-slip boundary conditions are satisfied and red spheres are beads of the polymer chain.

Schematic (top) and discretized representation (bottom) of the nanopit problem. Gray spheres indicate points at which no-slip boundary conditions are satisfied and red spheres are beads of the polymer chain.

(Top) Streamlines in the nanopit and (bottom) contour plot of the streamwise velocity on the *x* − *z* plane.

(Top) Streamlines in the nanopit and (bottom) contour plot of the streamwise velocity on the *x* − *z* plane.

Snapshots of a hopping event (from (a) to (f)) (top-down view, Pe = 3.5).

Snapshots of a hopping event (from (a) to (f)) (top-down view, Pe = 3.5).

(a) Time series of the *x*-component of the center-of-mass of a λ-DNA molecule driven through a nanopit array for two low values of Pe. (b) Residence time distribution and best fit to an exponential distribution for the conditions shown in (a).

(a) Time series of the *x*-component of the center-of-mass of a λ-DNA molecule driven through a nanopit array for two low values of Pe. (b) Residence time distribution and best fit to an exponential distribution for the conditions shown in (a).

Mean residence time τ vs. Péclet number Pe for simulations including and neglecting hydrodynamic interactions (labeled HI and FD, respectively).

Mean residence time τ vs. Péclet number Pe for simulations including and neglecting hydrodynamic interactions (labeled HI and FD, respectively).

Mean residence time τ vs. chain length *N* and Péclet number Pe. For the case Wi = 24, results with and without HI are shown.

Mean residence time τ vs. chain length *N* and Péclet number Pe. For the case Wi = 24, results with and without HI are shown.

(a) Residence time distribution at high Péclet number and best fit to a Gaussian distribution. (b) Parameters for the Gaussian residence time distribution as a function of Péclet number.

(a) Residence time distribution at high Péclet number and best fit to a Gaussian distribution. (b) Parameters for the Gaussian residence time distribution as a function of Péclet number.

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