Re: [ESPResSo-users] Brownian motion with lattice Boltzmann

[Ulf Schiller]

Hi,

On 03/02/2012 11:25 PM, Stefan Kesselheim wrote:
> Dear Salvador, dear Espresso-Users,
>
> Am 02.03.2012 um 17:36 schrieb Salvador H-V:
>
> > Dear All,
> >
> > I have been reading the Thesis of Dominic Røhm where he discuss the 
> > implementation of Lattice Boltzmann with GPU's in ESPResSo.
> >
> > In chapter 8.2, he presents a test case about brownian motion and diffusion of 
> > a single particle.
> >
> > Some of the parameters used in the simulation were:
> >
> > density           1.0
> > viscosity         0.1
> > friction            5.0
> > temperature    1.0
> >
> > I would like to ask for an educated guess of were this parameters come from? 
> > Somebody could give me an idea of why these parameters are good enough to 
> > reproduce brownian motion and if they could be used in a whole 2D colloidal 
> > suspension?

In general, three out of the four parameters fix your mass, length and 
time units. The fourth parameter, typically the friction, is used to fix 
the diffusion coefficient.

The parameters above correspond to a Schmidt number of roughly Sc ~ 0.5 
which I would expect to be low enough to see diffusion in a 2d colloidal 
suspension.

Note that technically the particles are point particles, whose apparent 
size is determined by the grid resolution. This may lead to problems in 
determining a hydrodynamic radius (typically defined in terms of hard 
spheres), as Stephan has pointed out. For colloids of finite size one 
could also use the raspberry model by Lobaskin and Duenweg.

Regards,
Ulf

> As far as I know, this parameters were chosen not for a particular purpose 
> beyond LB works stable with this set (missing is the more important LB 
> timestep, most likely 0.01, that actually determines the speed of sound and the 
> Boltzmann Number (cf. Duenweg/Ladd Review article).
>
> Brownian Motion can, in principle, be reproduced by any parameter sets, under which the 
> algorithm is stable, because the term "Brownian Motion" has no meaning beyond 
> random motion with a well defined temperature. The question is just: What will be the 
> Diffusion constant you get.
>
> Hidden in the Diffusion constant is the effective hydrodynamic particle radius: 
> One would define it as k_B T /6/pi/eta/D : The radius of a Stokes sphere with 
> the same Diffusion coefficient in the same solvent (eta is visc). If your 
> parameter set creates high Diffusion coefficients at high viscosity, your 
> particles become small (from a hydrodynamic point of view). This can reduce the 
> hydrodynamic interactions (\propto r_h) until the disappear fully. Early works 
> by Duenweg and Ahlrichs suggested that good parameter sets have hydrodynamic 
> radii ~ 0.5 lattice constant.
>
> A careful analysis of this is still missing in the literature, and we are currently 
> working on that (hopefully it will eventually come out). My suggestion for a good 
> parameter set would be agrid 1 density 1 viscosity 0.8 friction 10 tau 0.01 . This gave 
> good results for a 3D system of uncharged spheres and probably works in well in many 
> cases. The hydrodynamic radius is ~0.45 lattice constants then. To me it is not very 
> clear, why it actually is good and "superior" to other parameter sets, but it 
> might however be a good starting point.
>
> Sorry for not being more clear in the answers, but it is still research in 
> progress...
> Cheers and good luck
> Stefan
> PS: If you have any trouble with the LB implementation, please let us (mostly 
> Dominic and me) know.
>
>
> > Thanks a lot in advance,
> >
> > Herrera-Velarde S
> >
> >
> >
> > --
> > =o=o=o=o=o=o=o=o=o=o=o=o=o=o=o=o=o=o=o
> >
> > Dr. Salvador Herrera Velarde
> > Division de Ciencias e Ingenierias
> > Campus Leon
> > Universidad de Guanajuato
> >
> > =o=o=o=o=o=o=o=o=o=o=o=o=o=o=o=o=o=o=o
> >
> > Este correo ha sido editado para evitar el uso de acentos y guardar 
> > compatibilidad entre
> > diferentes distribuciones
>
> Stefan Kesselheim
> Institute for Computational Physics
> University of Stuttgart
> address@hidden

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