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Diel const from molecular dipole: catastrophe, Kirkwood (fwd)
Sender: bruce_bush@merck.com (Bruce Bush)
Subject: WSN: Diel const from molecular dipole: catastrophe, Kirkwood
Venkat Vasudevan writes of trying to get water dielectric constant
(eps) from molecular dipole moment and polarizability. The result is
infinite or negative. This is absurd.
> Seems like a trivial question but I am puzzled. I am using well known
> data on dipole moment (1.86 Debye) and polarizability volume (1.47 Ang^^3) of
> water to calculate its dielectric constant at 298 K.
> I am using the well known Debye equation, namely,
>
> (eps' -1)/(eps'+2) = Pm*rho/Mm where
>
> Pm = [Nav/(3*eps0)] * {alpha + mu**2/3kT} is the molar polarizabilty.
> rho is the density (g/cm3) and Mm is the molar mass in g/mol.
>
> Here eps' -dielectric constant, eps0- vac.permitivity= 8.854 E-12 C^^2/(J M)
> alpha is the polarizability, mu is the dipole moment and k is the Boltzman
> constant (1.38E-23 J/K). The unit of debye in SI units is 1 D = 3.336E-30 C
> Metre.
> : : : : : :
> (eps'-1)/(eps'+2) = 70.05 (cm3/mol)* 1.00 (g/cm3)/ 18.0 (g/mol)= 3.93.
>
> This would give an eps' value to be negative which is ridiculous. Where is
> the problem? I know it is somewhere in the conversion. But I checked a lot
> of times and it seems right. Can anyone spot the mistake? Thanks in advance.
> BTW, the dielectric constant for water is around 78.5 (eps').
>
The problem is not with "conversion"
but with the use of the 'Debye' formula (also called, I think, Lorentz-
Lorenz and Clausiu-Mossotti). The formula is only approximate and holds for
relatively weak dielectrics where the dielectric constant is near one.
Note that for eps -> +infinity the right hand side approaches 1.0; that is,
the formula diverges as (Pm) * (number density) goes to 1.0.
The Debye/C-M/L-L formula relates the average
local polarization to the field produced by all the surrounding average
polarizations. This is done self-consistently, so the effect (x) of the
local polariztion is added to the original field and in turn
produces response x**2, which produces x**3, and so forth.
This self-consistency leads to an infinite result as the
series 1+x+x**2+x**2+x**3 diverges. In an actual thermodynamic equilibrium
system this divergence does not occur. Jackson, Classical Electrodynamics, 1st
Ed., p. 119 alludes to the problem but goes no further.
In short: "epsilon" is a property of a bulk medium and can be defined by
macroscopic manipulations (applying a field in a capacitor and measuring
energy or charge on capacitor plates). This property is valid but
tricky to relate to a *molecular* model of the system. The Debye/C-M/L-L
formula applies in the limit where the chemical molecules (H2O) are
the appropriate "actors" from which the behavior of the entire system
can be built up. But there's nothing exact about such a formula; if the
liquid consists of highly correlated domains, then the domains become
the appropiate "actors" to plug into a Debye-like formula. The only
way to get numbers for the effective moment and density of these domains
is to do a simulation of the correlations between the molecules.
The accurate expression for (eps) involves the "Kirkwood g-factor"
expressing correlations between neighboring dipoles.
Recent papers in the simulation field discuss or evaluate the g-factor.
(A paper by et al. and Honig, probably Gilson et al., is an example.)
Allen & Tildesley (Computer Simulation of Liquids, Oxford, p. 161) say that
one can simulate a box of waters surrounded by a medium of dielectric ES;
if the surround is a vacuum, for example, ES=1. Keep track of the fluctuations
in the *total* dipole moment of the system, which incorporates the correlations:
g(ES) = (1/N.mu**2) { < ( Sum(_mu_) )**2 > - ( < Sum(_mu_) ) > **2 }
where _mu_ is a vector; <...> denotes ensemble average.
So the right hand side gives the variance or fluctuation of the
total Sum(_mu_) about its mean or average. Then
(eps - 1) / (eps + 2) = g(1)*y
where (y) is the quantity (Pm*rho/Mm) appearing in the uncorrected formula.
Allen and Tildesley do not recommend using this formula, as it converges
slowly. In general, they say, for any choice of ES
(1/(eps-1)) = (1/(3yg(ES))) - (1/(2ES+1)).
The resulting value of (eps) is independent of ES.
If one knew (eps) then the best choice for the surrounding ES would be (eps)
which would lead to "the Kirkwood formula [1949]"
(2*eps+1)*(eps-1)/(9*eps) = yg(eps).
Since we don't know (eps), but suspect it is large, they recommend using
the formula for ES=infinity:
eps = 1 + 3y(g(ES=inf))
(Note that this has no way of blowing up). The way to get g(ES=inf) is
apparently to do a simulation in vacuum but to assume that the "surface
charges" (where the dipoles of the system meet the vacuum) don't exist.
Because Coulomb forces are long-range, A&T indicate that one should
calculate the interactions for the simulation using "Ewald sums": that is,
replicate a central box of waters periodically, and use a trick to allow
the sum over all boxes to be done almost as fast as just the sum over
the central waters.
Perhaps someone who has actually done such simulations, or experiments
that probe g() directly, can give you more information.
-- Bruce
Bruce_Bush@merck.com (908) 594-6758
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