# Molecular Dynamics and Stokesian Dynamics simulations

## Stokesian Dynamics simulations of sedimenting particles in a viscous liquid

The dynamics of particles which move in a highly viscous liquid such all motions are overdamped and random forces can be neglected is called Stokesian Dynamics. The simplest example is a macroscopic sphere which moves in oil under the influence of a constant force like the gravitional force. According to the famous Stokes law the resulting velocity is proportional to the driving force. Other examples for which the Stokesian dynamics description is appropriate are vesicles, blood cells, many colloidal suspensions, and large proteins which are driven by strong forces as they occur in ultracentrifugation.

To describe the dynamics of many-particle systems one needs to take into account hydrodynamic interactions (HI) which are mediated by the liquid. Mathematically they are described by the friction matrix which replaces the friction constant appearing in the Stokes law for a single sphere. HI are long-ranged many-body effects. The latter means that the HI between two particles is modified by the presence of a third one which is nearby.

### 9 equally sized spheres starting on a quadratic grid sedi_5sphere_name.mov (FileTypequicktime, 15 MB)

1.a) The sedimentation of 9 spheres in a viscous liquid is shown. The center-of-mass motion is subtracted. All spheres are identical. The colours label symmetrically equivalent spheres. The little dots on the spheres serve to indicate rotational motion. The falling black bar on the lefthand side shows the sedimentation of the whole cluster. Its size corresponds approximately to the part of the system which is shown.

### 9 equally sized spheres starting on a quadratic grid

(periodic boundary conditions, box = 8 diameters, center-of-mass motion substracted) sedi_plane_name.mov (FileTypequicktime, 33 MB)

1.b) Compared to 1.a) periodic boundary conditions are applied which would be normally used to simulate infinite systems. The size of the replicated cubic box is 8 sphere diameters. Here the effect of HI interactions between the periodic images is shown which leads essentially to more friction and therefore to a slowing down of the sedimentation. The local motions are only slightly influenced.

### 9 equally sized spheres starting on a slightly perturbed quadratic grid

(center-of-mass motion substracted) sedi_plane_pcb_name.mov (FileTypequicktime, 35 MB)

1.c) Here the influence of a slight perturbation of the initial configuration is shown. The spheres in the start configuration are randomly displaced by 10% of the radius of a sphere. Note that the red sphere forms a dimer with a green sphere. This effect is called 'kissing' and is only due to hydrodynamic interactions.

### Pentamer starting in the stretched conformation

(center-of-mass motion subtracted) sedi_poly5_name.mov (FileTypequicktime, 35 MB)

1.d) The movie shows the sedimentation of a pentamer which can be thought as simple model for a large molecule consisting of five domains ('monomers') which are interconnected by hinges. In the simulation these hinges are joints with three rotational degrees of freedom which are placed halfway between the spheres. They do not influence the motion of the latter.

### 5 equally sized spheres starting in a linear configuration

(center-of-mass motion subtracted) sedi_5sphere_name.mov (FileTypequicktime, 15 MB)

1.e) This is a reference simulation for 1.d) in which the constraints are removed.

The movies were generated with molscript and raster3d.