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

[Markus Deserno]

Folks,

quite interesting into what type of good discussions one can get by contemplating

a single physical concept. It seems we're converging, but let me add a few more

short points:

> You say, that the Bjerrum length is the distance where the potential of mean force

> of two ions in water equals k_B T. But this would mean, that the Bjerrum length ion

> (pair) specific. Would that be exactly what we want?

This is a good point. I didn't want to go into this, but since you bring it up, I should

explain this.

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.

> 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.

> 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!

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.

Best,

Markus Deserno

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On Oct 6, 2013, at 7:37 AM, Ulf Schiller <address@hidden> wrote:

On 10/06/2013 11:22 AM, Peter Košovan wrote:

Similar trouble arises when we use point charges but replace the
continuum solvent with an explicit one. We can measure the dielectric
constant epsilon_model of our solvent model. This will probably differ
from the experimental value epsilon_exp since the model is not exact.

To my understanding, the solution is to adjust our bjerrum parameter
such that
1/epsilon_exp = 1/(epsilon_model * epsilon_bjerrum) where
epsilon_bjerrum is obtained by substituting the value of bjerrum
parameter we provided to Espresso into the common formula for Bjerrum
length.

To this end, I do not see inconsistency between what Stefan and Marcus
have written.

Btw: does anyone know how users of other software deal with the problem
that epsilon_model does not match epsilon_exp? Do they use corrections
to epsilon similar to what I described or simply ignore the difference?

Hi Peter,

I think this is a software-independent modeling question. The Bjerrum length would be just one of potentially many input parameters of an explicit water model. If you take a given model and adjust only the Bjerrum length to match epsilon, it is not unlikely that you 'destroy' some other property like the diffusivity. Basically, the whole set of input parameters among with the degrees of freedom of the model have to be tuned to get the right physics. If I remember correctly, some SPC water models are able to reproduce the dielectric constant.

Best,
Ulf

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