^{1,a)}and J.-U. Sommer

^{1,2,3,b)}

### Abstract

We consider molecular dynamics simulations of the electrophoretic motion of charged polymers in straight cylinders and cylinders with a periodic variation (or modulation) of the diameter. The electric field is always orientated along the axis of the cylinders, and it does not change direction. The fluid is modeled explicitly, also the co-ions, counterions, and the charged polymer monomers. In straight geometries, we observe no separation effect for a wide range of applied electric fields. In modulated geometries, the charged polymer chains can be separated only at low fields. At large fields, we observe separation effects in modulated geometries only if the applied field is a dc pulse electric field. A simple scaling theory to explain the observed behavior is presented.

I. INTRODUCTION

II. METHOD

III. RESULTS AND DISCUSSIONS

IV. SCALING THEORY

V. CONCLUSIONS

### Key Topics

- Polymers
- 34.0
- Electrophoresis
- 25.0
- Electric fields
- 23.0
- Hydrodynamics
- 11.0
- Diffusion
- 10.0

## Figures

(Top) Schematic cross-section of (dashed) constant, and (black) periodically modulated diameter cylinders. (Middle) The corresponding surface gradient *R* ^{′}(*z*) as a function of z-coordinate. (Bottom) The driving pulse electric field of maximum amplitude as a function of time. The off-time window takes 70% of a time period *T* = 578τ. δ notes the diameter of the constriction.

(Top) Schematic cross-section of (dashed) constant, and (black) periodically modulated diameter cylinders. (Middle) The corresponding surface gradient *R* ^{′}(*z*) as a function of z-coordinate. (Bottom) The driving pulse electric field of maximum amplitude as a function of time. The off-time window takes 70% of a time period *T* = 578τ. δ notes the diameter of the constriction.

Illustration of a sample configuration from molecular dynamics simulations of (black) charged polymer chain of length N = 150, (green) coions and (red) counterions in confining cylinder of variable diameter. The chain extends on two neighboring cavities and passes through constrictions by forming loops. (The direction of motion is from right to left.) For clarity, the solvent molecules are not represented here.

Illustration of a sample configuration from molecular dynamics simulations of (black) charged polymer chain of length N = 150, (green) coions and (red) counterions in confining cylinder of variable diameter. The chain extends on two neighboring cavities and passes through constrictions by forming loops. (The direction of motion is from right to left.) For clarity, the solvent molecules are not represented here.

Electrophoretic velocities *v* _{ z } of charged chains of length *N* as a function of the applied field in uniform (noted cyl.) and modulated (noted mod.) diameter cylinders, and in modulated cylinders under pulse field of amplitude (noted _{−}⊓_{−}). The error bars are smaller than the symbol size. (The off-time window of the pulse field is 70% of a time period.)

Electrophoretic velocities *v* _{ z } of charged chains of length *N* as a function of the applied field in uniform (noted cyl.) and modulated (noted mod.) diameter cylinders, and in modulated cylinders under pulse field of amplitude (noted _{−}⊓_{−}). The error bars are smaller than the symbol size. (The off-time window of the pulse field is 70% of a time period.)

Bar diagram of average positions *z* after a set time interval of electrophoresis of charged chains of length N in constant (noted cyl.) and variable diameter (noted mod.) cylinders under constant applied fields , and similar positions in modulated cylinders (noted _{−}⊓_{−}) in applied pulse fields of relative strength

Bar diagram of average positions *z* after a set time interval of electrophoresis of charged chains of length N in constant (noted cyl.) and variable diameter (noted mod.) cylinders under constant applied fields , and similar positions in modulated cylinders (noted _{−}⊓_{−}) in applied pulse fields of relative strength

Schematic illustration on the center of mass motion *z* = *z*(*t*), at constant low fields , of the trapping time *t* _{ b } of charged chains in the cavities of modulated diameter cylinders and the transit velocity *v* _{ t } through constrictions.

Schematic illustration on the center of mass motion *z* = *z*(*t*), at constant low fields , of the trapping time *t* _{ b } of charged chains in the cavities of modulated diameter cylinders and the transit velocity *v* _{ t } through constrictions.

Illustrative trajectories at small time scale of the center of mass motion of charged chains in modulated diameter cylinders under pulse electric fields. The relaxation (or free diffusion) of the chains in the off-time window of the pulse field appears as a backtrack motion and makes the trajectories appear zigzagged.

Illustrative trajectories at small time scale of the center of mass motion of charged chains in modulated diameter cylinders under pulse electric fields. The relaxation (or free diffusion) of the chains in the off-time window of the pulse field appears as a backtrack motion and makes the trajectories appear zigzagged.

The long time average of the transverse radius of the chain normalized by the bond length *b* as a function of constant applied field in constant (noted cyl.), variable diameter cylinders (noted mod.), and in pulsed fields (noted _{−}⊓_{−}). The error bars are smaller than the symbol size. (In the case of pulse fields, is rescaled by a factor of 1/3 to account for the off-time window of a time period.)

The long time average of the transverse radius of the chain normalized by the bond length *b* as a function of constant applied field in constant (noted cyl.), variable diameter cylinders (noted mod.), and in pulsed fields (noted _{−}⊓_{−}). The error bars are smaller than the symbol size. (In the case of pulse fields, is rescaled by a factor of 1/3 to account for the off-time window of a time period.)

Projected length |*z* _{ max } − *z* _{ min }| in the field direction normalized by the contour length *Nb* of the chains. The legend notation is the same as in Fig. 7 . The error bars are smaller than the symbol size.

Projected length |*z* _{ max } − *z* _{ min }| in the field direction normalized by the contour length *Nb* of the chains. The legend notation is the same as in Fig. 7 . The error bars are smaller than the symbol size.

Transit velocities *v* _{ t } and average electrophoretic velocities *v* _{ z } of the charged chains driven in modulated diameter cylinders as a function of the applied field (not pulse field). At high fields, both *v* _{ t } and *v* _{ z } coincide. The largest chain of *N* = 150 passes through constrictions almost unhindered, at fields .

Transit velocities *v* _{ t } and average electrophoretic velocities *v* _{ z } of the charged chains driven in modulated diameter cylinders as a function of the applied field (not pulse field). At high fields, both *v* _{ t } and *v* _{ z } coincide. The largest chain of *N* = 150 passes through constrictions almost unhindered, at fields .

(Left) Schematic representation of a compression blob model of a chain squeezed in a cylindrical confining cavity of diameter *R* _{0}. At equilibrium between forces of entropic and electric origin, the chain occupies the confined volume . (Right) At higher applied electric fields, *n* monomers of the chain are pushed into the connecting channel of diameter δ for a distance *x*.

(Left) Schematic representation of a compression blob model of a chain squeezed in a cylindrical confining cavity of diameter *R* _{0}. At equilibrium between forces of entropic and electric origin, the chain occupies the confined volume . (Right) At higher applied electric fields, *n* monomers of the chain are pushed into the connecting channel of diameter δ for a distance *x*.

(Points) Log-linear plot of the trapping time *t* _{ b } of the chains in the cavities of the variable diameter cylinder as a function of the applied electric field (not pulse) and (continuous line) scaling prediction from Eq. (16) . Only at the lowest field value *t* _{ b } could be measured for the chain with *N* = 150.

(Points) Log-linear plot of the trapping time *t* _{ b } of the chains in the cavities of the variable diameter cylinder as a function of the applied electric field (not pulse) and (continuous line) scaling prediction from Eq. (16) . Only at the lowest field value *t* _{ b } could be measured for the chain with *N* = 150.

Electrophoretic mobilities μ_{ z } of the chains in straight and variable diameter cylinders as a function of the applied electric field.

Electrophoretic mobilities μ_{ z } of the chains in straight and variable diameter cylinders as a function of the applied electric field.

Electrophoretic mobilities μ_{ z } of the chains under applied pulse electric fields in variable diameter cylinders. ( is rescaled by a factor of 1/3 to account for the off-time window of the time period.) As a reference, the electrophoretic mobilities μ_{ z } of the chains under applied constant electric fields from Fig. 12 are also reproduced here.

Electrophoretic mobilities μ_{ z } of the chains under applied pulse electric fields in variable diameter cylinders. ( is rescaled by a factor of 1/3 to account for the off-time window of the time period.) As a reference, the electrophoretic mobilities μ_{ z } of the chains under applied constant electric fields from Fig. 12 are also reproduced here.

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