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Re: [ESPResSo-users] Bjerrum Length for Explicit Solvent

From: Stefan Kesselheim
Subject: Re: [ESPResSo-users] Bjerrum Length for Explicit Solvent
Date: Sun, 6 Oct 2013 16:14:41 +0200

Dear Marcus, dear rest,

only a few brief comments...

On Oct 6, 2013, at 3:22 PM, Markus Deserno <address@hidden> wrote:

> At sufficiently large distance the electric field D(r) away from a unit 
> charge is not
> specifically dependent on the charge: it assumes the value q/(4 pi eps_0 
> eps_r r^2),
> and the dielectric constant eps_r is a material property independent of that 
> ion.
> The problem is, of course, that as you get closer to the charge, it has a 
> finite radius
> and some funny distribution of electrostatic orbitals. Hence, at some point 
> the
> electrostatic field will no longer be inversely proportional to 1/r^2. If the 
> distance
> at which the electrostatic contribution to the free energy of interaction 
> gets so close
> to the ion that it begins to approach this region of specificity, you can of 
> course still
> determine that distance, but it has non-electrostatic effects entangled into 
> it which
> are hard if not impossible to disentangle. I agree with you that this is a 
> limitation
> of the concept. I can of course still go ahead and define the Bjerrum length 
> in such
> a way that the asymptotic field is correct, which essentially means defining 
> it via
> the bulk dielectric constant, but I concede that at that moment one is 
> looking for
> workarounds. The fact that unit charges are not point charges leaves an ugly 
> trace.

You are right: Asymptotically the electrostatic interaction of two charges of 
whatever type (point, atoms, …) approaches Coulomb's law with the bulk 
dielectric constant inside. This is just related to the fact that all other 
interactions presumably decay faster electrostatics. (What about at a critical 
point though …?) You could also say: The prefactor of the asymptotic force 
between two point charges is defined as the dielectric permittivity (up to 4 
pi, or such stuff). This would include a measurement principle and hence be a 
physical definition of a quantity. For the temperature, there are protocols to 
measure it.

> > In my opinion the Bjerrum length can not be measured, and is not a physical 
> > quantity.
> > It is a theoretical concept that is used to combine measurements of the 
> > bulk (!) dielectric
> > permittivity and the temperature of a medium into a single quantity that is 
> > helpful to
> > keep formulas brief. 
> One can take that point of view, but of course the Bjerrum length is then 
> still measurable,
> since I can measure the bulk dielectric constant. And I'd like to point out 
> that as long as
> the local packing effects are small, the potential of mean force is a valid 
> approximation.
> Unfortunately the case of water is really pushing this limit, since the 
> Bjerrum length is
> so small. But in solvents with a lower dielectric constant I expect the 
> discrepancies to
> become bigger.

Did you mean the last sentence as written? In lower dielectric constant it 
should become less problematic, right? The Bjerrum length is then larger than 
the molecular size, and the "granularity" of the medium is less important. 

But my point was: The Bjerrum length _not_ measured by taking two charges and 
pushing them together until the work you have spent is k_B T. Its definition is 
stems from the asymptotic behaviour, and that is by definition the dielectric 

> > Both definitions are possible, but not compatible. The first is a (bulk) 
> > material property
> > at a particular temperature, the second is a property of ion pairs in a 
> > particular medium
> > at a particular temperature.
> As far as compatibility is concerned, the only issue is the finite size of 
> the ions, which
> contributes extra terms to the potential of mean force, and that can be nasty 
> and unfortunate.
> However, the fact that the Bjerrum length would be medium and temperature 
> dependent
> is no problem. Of course it depends on that. It's a prefactor in a  free 
> energy!

It is important to stress here that the "free energy" because this means that 
is invalid to simulate two charges in water for different distances, calculate 
the mean electrostatic energy energy for all distances and use this energy for 
the definition of the Bjerrum length. Ulf's suggestion could be understood like 
that. The thing that we work with in implicit solvent simulations is not an 
energy as for an atomistic system, but it is a Free Energy where all implicit 
degrees of freedom have been integrated out. 

> Let's please not forget why I started this discussion: All I wanted to make 
> sure is that
> we understand that the Bjerrum length is a physical concept that is 
> independent of
> modeling. If one wants to model water, then the physics tells us what the 
> Bjerrum length
> ought to be at the end of the day. If you would like to use the theorists 
> view that it is
> just a convenient way of writing a bulk Poisson equation, I'm all with you, 
> but still,
> that length is then physical and is characteristic of the solvent, just like 
> the dielectric
> constant is, and it will also depend on temperature. Hence, the physical 
> quantity Bjerrum
> length ought to be distinguished from an input parameter in some Espresso 
> script.

That sounds fair to me, as long as we make clear: Every microscopic model has 
its own Bjerrum length, for a given temperature. It depends all on the 
microscopic model how large it is. 

Cheers and have a nice sunday!

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