On Nov 4, 2009, at 1:26 AM, Michel Jullian wrote:
2009/11/4 Horace Heffner <[email protected]>:
As you certainly know the loading factor (hydrogen content)
depends on
the current density. If you have more current density on one
side, you
will have more hydrogen content on that side. And then by diffusion,
you will have migration to the other side and desorption.
Implacable,
it's like trying to inflate a punctured tyre.
I'd like to see a reference on that relationship. I don't think
there is a
linear relationship between loading flux and hydrogen evolution or
current
density.
Then you don't think right ;-)
In the case of single side electrolytic loading of Pd membranes (with
the other side unloaded),
I assume by "unloaded" you mean there was no electrolyte interface on
the far side. That is essentially the same situation as a crack.
unpublished research I have participated in,
based on experimental data from various sources, does show a nice
linear relationship between steady state permeated flux through the
membrane and current density, for current densities above 10mA/cm^2.
The proportionality factor is inversely proportional to thickness,
apart from this it only depends on the cathode temperature (increases
by a factor 1.6 for every 20°C increase), and on the isotope
(permeation for D is higher than that for H). Here are the empirical
formulae we obtained:
H2: Permeated flux (mL hr^-1 cm^-2) = 4.6 / thickness (µm) * current
density (mA cm^-2) *
1.6^((Tcath(°C)-45)/20)
D2: Permeated flux (mL hr^-1 cm^-2) = 7.4 / thickness (µm) * current
density (mA cm^-2) *
1.6^((Tcath(°C)-45)/20)
That is interesting data!
Here is a start at looking at this theoretically.
Looking at just the diffusion part of the picture, the Arrhenius Law,
for T=230 K to 1220 K, gives the diffusion coefficient D (in cm^2/s) as:
D = D0 * exp(-E/(k_b T))
where D is in (Kb T) is the thermal energy, E is called the
activation energy, and the values for Pd are given in Table 1 below.
Isotope, D0 (cm^2/s), E (eV)
H, 2.9x10^-3, 0.23 eV
D, 1.7x10^-3, 0.206 eV
T, 7.3x10^-4, 0.185 eV
Table 1 - Arrhenius Law Coefficients for hydrogen in Pd
Hydrogen flux, j(r,t), r being a direction vector, t time, is given by:
j(r,t) = -D del n(r,t)
in a concentration gradient, where n(r,t) is the hydrogen density as
a function of the space vector r and t.
Given 1 eV = 11,600 K, we have k_b above is
k_b = ((1 eV)/11600 K) = 8.621x10^-5 eV/K
Using temperature = 45 °k, we have T = 273.15 + 45 = 318.15 K, and
k_b T = 0.027427 eV.
For hydrogen at 45 °C we have:
D_h = (2.9x10^-3 cm^2/s) * exp(-0.23/0.027427) = 6.6x10^-7 cm^2/s
For deuterium at 45 °C we have:
D_d = (1.7x10^-3 cm^2/s) * exp(-0.206/0.027427) = 9.3x10^-7 cm^2/s
and the concentration based flux ratio should be D_h / D_d =
(6.6x10^-7 cm^2/s)/(9.3x10^-7 cm^2/s) = 0.71.
Your flux ratio at 45 °C is 4.6/7.4 = 0.621, which is not far in
disagreement, relatively speaking, at that temperature.
One problem with all this is cracks and surface features can and do
throw everything off as far as load achieved, and the energy required
for loading changes substantially between phases.
Let's look at things at 100 °C.
For hydrogen at 100 °C we have, k_B T = 0.0322 eV:
D_h = (2.9x10^-3 cm^2/s) * exp(-0.23/(8.621E-5 * (273.15 + 100))
= 2.28x10^-6 cm^2/s
For deuterium at 100 °C we have:
D_d = (1.7x10^-3 cm^2/s) * exp(-0.206/(8.621E-5 * (273.15 + 100))
= 2.8x10^-6 cm^2/s
and a concentration based flux ratio of 0.814 for H vs D.
Your formula still provides a comparable ratio of:
Ratio = (4.6 * 1.6^(55/20)) / (7.4 * 1.6^(55/20)) = 4.6/7.4 = 0.621
In other words, assuming loading is a function of current, i.e.
boundary concentration is a function of current density, and
diffusion flux matches the well established Arrhenius Law, it looks
like your data should show differing sensitivities of D and H to
temperature. Based on diffusion resistance only, at 45 °C, hydrogen
should move about 71 percent as fast as deuterium, while at 100 °C
hydrogen should move about 81.4 percent as fast as deuterium. Your
equations show hydrogen moving about 62.1 percent as fast as
deuterium in both cases. This means loading can not be a fixed
percentage of current density if the form of your equations is correct.
If you developed your equations doing regression analysis it might be
worth looking at changing the exponential form to match the Arrhenius
version just to see how it fits.
The above formulas were taken from "Hydrogen in Metals III", pp 52-63.
Now if there are different non null current densites on both sides of
a membrane, I speculate that the flux will be proportional to the
difference between the front and back side current densities, but I
know of no experimental data on this.
It appears it should indeed be feasible to develop a formula if it is
known what loading is achieved at a given potential. This would give
the expected concentration at the boundaries, and the rest is just a
matter of a diffusion coefficient. I think steady state loading as
a function of potential is highly non-linear, which means boundary
concentration is highly non-linear with respect to current, even
ignoring problems of surface features. It is also notable that if
potential drops then current density drops and thus temperature
drops, reducing the flow, the diffusion flux, exponentially. This is
a complicated model.
It is admittedly interesting that diffusion rates are exponential in
T, and the spots show up in the hot area. It is also true that
cavitation starts in hot spots.
Fusion or cavitation makes the sound? I'll still bet on cavitation.
Interesting that cavitation can be expected to create hot fusion, and
neutrons.
Michel
P.S. your comments below may be correct though, I haven't studied the
relationship between permeated flux and overpotential (which I don't
think is linear with current density)
Hydrogen evolution in the form of bubbles is almost entirely the
result of electronation, the tunneling of electrons across the
interface.
The tunneling probability is exponential in terms of the
overpotential.
When hydrogen is removed from the electrolyte side of the
interface it
drops the potential there. The solvated hydrogen concentration is
dropped
up close to the interface. The path for protons through the
interface to
the Pd is more linear in nature (compared to the electrons path in
the
opposed direction across the interface) with the overpotential
because it
requires a couple short distance proton tunnelings, and two water
molecule
rotations which are more linear with overpotential.
The metal side of the interface is clearly very close to the same
potential
everywhere. There ion conduction path to the back side covers
many routes,
but in the case of the mesh electrode is primarily through the
mesh, a mm or
less. In either case, the back side, or front side, the ion
conduction rate
is largely a product of diffusion and not potential gradient, at
least in
ordinary electrolytic cells, because most of the potential drop
and vastly
more gradient is at the interfaces. The field gradient between
electrodes is
so small that concentration gradients have more effect on the
diffusion
rates. However, it is unfortunately not clear at all what kinds of
potentials and currents are involved. We are left to guess what is
happening
As you can see, I have a very difficult time seeing that such a
small change
in potential can be responsible for that much "fusion". I may be
highly
biased though, especially by my work with anode discharges, which
sound and
look very similar, except in the visual range.
I find it concerning that the sound track on the film sounds very
much like
an electrospark experiment. It sounds like cavitation. The noise
shows up
when the cathode heats up, just like electrospark. I eventually
started
heating up my cells to boiling before starting (I was running a
boiloff
protocol) just to get more uniform and faster data. I think the
cathode
white dots may in part be steam bubbles.
Best regards,
Horace Heffner
http://www.mtaonline.net/~hheffner/
