Excerpt from Thesis :
Characterising Porous Media.
John Beausire Wyatt Webber,
Ph.D. Thesis, University of Kent, UK, 2000.
On the Web :
http://www.kent.ac.uk/physical-sciences/publications/theses/jbww.html
Chapter 3 NMR Cryoporometry method.
NMR Cryoporometry is a method suitable for measuring pore sizes and
pore
size distributions in the range of less than 30Å to over 3000Å
pore diameter. The technique involves freezing a liquid in the
pores and measuring the melting temperature by Nuclear Magnetic
Resonance. Since the melting point is depressed for crystals of
small size, the melting point depression gives a measurement of
pore size.
3.1 NMR Cryoporometry equations.
We may define porosity as the fraction of the sample volume
occupied by pore void; an important characterising parameter is
the porosity as a function of pore diameter and spatial location,
i.e. the pore size distribution within a material.
Recent work by Jackson and McKenna
demonstrated that the
melting point depression of a variety of liquids confined in
silica glasses varied inversely with the mean pore diameter as
determined by gas adsorption isotherm measurements.
This behaviour is closely related to the capillary effect; both
reflect the change in free energy caused by the curvature of the
interfacial surface.
This behaviour was originally described by equations
developed by J.W. Gibbs,
based on theoretical considerations of the equilibrium states
of heterogeneous substances.
Initial impetus to this field of research was given by J. Thomson
who considered the related theory of the depression of melting-point
of ice caused by pressure or other stress
Sir W. Thomson (his brother, later Lord Kelvin)
derived a related theory for the equilibrium pressure at curved
liquid/vapour surfaces, that describes
capillary condensation (the Kelvin equation, see
chapter 1.3.1).
J.J. Thomson later considered the effect of curvature on the
equilibrium temperature of a liquid droplet
When extended to small crystals
this is consistent with the Gibbs and
Kelvin equations; the contact angle
is assumed to be 180°.
Expanding and taking the first term gives us the standard form.
Thus the Gibbs-Thomson equation for the melting point depression
ΔTm for a small crystal of diameter x
is given by
where : |
|
|
Tm |
= |
normal melting point of bulk liquid |
Tm(x) |
= |
melting point of crystals of diameter x |
σsl |
= |
surface energy at the liquid-solid interface |
ΔHf |
= |
bulk enthalpy of fusion (per gram of material) |
ρs |
= |
density of the solid |
We may rewrite equation
3.1 as :
To exploit this effect for pore size measurement
Ref:Strange+Rahman+Smith
a porous sample
containing a liquid is cooled until all the liquid is frozen, and
then gradually warmed while monitoring the amplitude of the NMR
proton spin echo from any liquid present. The liquid is usually
chosen to be water or cyclohexane, the latter offering the large
melting point depression factor k of 1825 KÅ.
i.e. a depression of nearly 20K in 100Å pores.
NMR is a sensitive technique for distinguishing between solid and
liquid, as the coherent transverse nuclear spin magnetisation
decays much more rapidly in a solid than in a mobile liquid
(chapter 2.7).
Measurement of the volume of liquid present is usually most
conveniently made using a
90 °x− τ−180 °y− τ−echo sequence
(chapter 2.8),
where the time interval 2τ is set to be longer than the solid
decay time but less than the decay time in the liquid
(chapter 7.1).
For
water and cyclohexane 2τ times of 4ms to 40ms were typically used.
The amplitude V of the echo is related to the volume v
of solid that has
melted to a liquid at a particular temperature T and thus the
volume of the pores that have dimension less than or equal to the
corresponding dimension x in equation
3.2.
A further small
increase in temperature T produces a small increase in liquid
volume Δv proportional to the volume of pores with diameter x to
x + Δx . The pore size distribution function dv/dx can therefore
be obtained from the slope of the curve of v vs T using :
where x is related to the temperature by the equation
3.2. The
value of k used was determined using gas adsorption data,
using the nominal pore sizes of a number of sol gel silicas
as calibration values
(chapters 7.5.3, 7.4.3),
Further calibration work was performed using small angle neutron
scattering
(chapters 11 to 17),
in an attempt to establish an absolute calibration scale.
Good agreement was established between the two methods for large pores,
but a significant divergence was found for small pores.
Thus the calculated pore size distribution gives one in effect the
incremental volume of the pores at a particular pore diameter,
for unit increment of pore diameter.
If one normalises the distribution to unit
volume of the dry porous matrix,
one obtains the units {Å−1}.
If one normalises the distribution to unit
mass of the dry porous matrix,
one obtains the units {l·Å−1·g−1}.
Thus if one then integrates these pore size
distributions over the measured pore diameter range, one obtains
respectively
fv , the volume fraction of the matrix occupied by void space,
and
pm , the mass normalised total porosity of the pores
{l·g−1}.
There has been considerable work by both NMR and neutron scattering
to understand the changes that occur in water/ice and cyclohexane
when enclosed in small pores