^{1,a)}, Alvaro Domínguez

^{2}, Matthias Krüger

^{3}and Florencia Penna

^{4}

### Abstract

We present a *dynamic density functional theory* (dDFT) which takes into account the advection of the particles by a flowing solvent. For potential flows, we can use the same closure as in the absence of solventflow. The structure of the resulting advected dDFT suggests that it could be used for nonpotential flows as well. We apply this dDFT to Brownian particles (e.g., polymer coils) in a solvent flowing around a spherical obstacle (e.g., a colloid) and compare the results with direct simulations of the underlying Brownian dynamics. Although numerical limitations do not allow for an accurate quantitative check of the advected dDFT both show the same qualitative features. In contrast to previous works which neglected the deformation of the flow by the obstacle, we find that the bow wave in the density distribution of particles in front of the obstacle as well as the wake behind it are reduced dramatically. As a consequence, the friction force exerted by the (polymer) particles on the colloid can be reduced drastically.

The authors thank S. Dietrich for financial support and fruitful discussions. A.D. acknowledges financial support from the Junta de Andalucía (Spain) through the program “Retorno de Investigadores.” M.R. acknowledges funding by the Deutsche Forschungsgemeinschaft within the priority program SPP 1164 “Micro- and Nanofluidics” under Grant No. RA 1061/2-1.

I. INTRODUCTION

II. ADVECTED dDFT

III. EXAMPLES

A. Ideal polymers

B. Gaussian polymers

C. Drag force on colloids

IV. CONCLUSIONS

### Key Topics

- Colloidal systems
- 45.0
- Polymers
- 31.0
- Solvents
- 29.0
- Stokes flows
- 28.0
- Polymer solution flows
- 13.0

## Figures

(Color) Cross section of the flow field given in Eq. (1) in a plane parallel to the direction of motion around (a) a small spherical colloid and (b) a big one (full circles, radius ). The dashed circles of radius mark the points of closest approach of solute’s centers (point in the center of open circles) to the colloid. The solute’s diameter in its mutual interaction is and, for nonadditive mixtures, not necessarily equal to its diameter (indicated by the dotted circle) in the interaction with the colloid. The component of the flow field normal to the dashed circle is larger for the small colloid (a) than for the large colloid (b). and are the same in both figures.

(Color) Cross section of the flow field given in Eq. (1) in a plane parallel to the direction of motion around (a) a small spherical colloid and (b) a big one (full circles, radius ). The dashed circles of radius mark the points of closest approach of solute’s centers (point in the center of open circles) to the colloid. The solute’s diameter in its mutual interaction is and, for nonadditive mixtures, not necessarily equal to its diameter (indicated by the dotted circle) in the interaction with the colloid. The component of the flow field normal to the dashed circle is larger for the small colloid (a) than for the large colloid (b). and are the same in both figures.

Contour plots of the density of ideal polymers for a flow velocity . The white circle at the origin is the colloidal particle with radius , the black circle is the annulus of thickness , and outer diameter which is unaccessible to the polymer centers due to the hard-wall interaction, see Fig. 1. corresponds to a uniform flow (i.e., to ). The maximum density in front of the colloid is . corresponds to . The bow-wave effect is reduced drastically. The maximum density in front of the colloid is .

Contour plots of the density of ideal polymers for a flow velocity . The white circle at the origin is the colloidal particle with radius , the black circle is the annulus of thickness , and outer diameter which is unaccessible to the polymer centers due to the hard-wall interaction, see Fig. 1. corresponds to a uniform flow (i.e., to ). The maximum density in front of the colloid is . corresponds to . The bow-wave effect is reduced drastically. The maximum density in front of the colloid is .

Density of ideal polymers at the point , , i.e., right in front of the forbidden zone around the colloid. (a) shows as a function of the colloid size for and . (b) shows as a function of for different values of .

Density of ideal polymers at the point , , i.e., right in front of the forbidden zone around the colloid. (a) shows as a function of the colloid size for and . (b) shows as a function of for different values of .

Plots of defined in Eq. (25) as provided by the numerical solution of dDFT (lines) and as measured in BD simulations (symbols). (a) corresponds to a velocity and (b) to . In each plot the results for both uniform flow and Stokes flow are presented.

Plots of defined in Eq. (25) as provided by the numerical solution of dDFT (lines) and as measured in BD simulations (symbols). (a) corresponds to a velocity and (b) to . In each plot the results for both uniform flow and Stokes flow are presented.

The plot represents , see Eq. (26), in a Stokes flow with upstream (at , triangles up) and downstream (at triangles down) of the colloid, as measured in BD simulations. The inset shows the results for a smaller simulation box.

The plot represents , see Eq. (26), in a Stokes flow with upstream (at , triangles up) and downstream (at triangles down) of the colloid, as measured in BD simulations. The inset shows the results for a smaller simulation box.

## Tables

Mean force exerted by the polymers measured in the BD simulations for different types of flow ( for the uniform flow, and for the Stokes flow), compared to the force exerted by ideal polymers and to the Stokes friction of the colloid. The forces are given in units of .

Mean force exerted by the polymers measured in the BD simulations for different types of flow ( for the uniform flow, and for the Stokes flow), compared to the force exerted by ideal polymers and to the Stokes friction of the colloid. The forces are given in units of .

Mean and variance of the force exerted by the polymers in a Stokes flow measured in BD simulations of different boxsizes (see Sec. III B). The forces are given in units of .

Mean and variance of the force exerted by the polymers in a Stokes flow measured in BD simulations of different boxsizes (see Sec. III B). The forces are given in units of .

Article metrics loading...

Full text loading...

Commenting has been disabled for this content