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 30 to over 3000 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
ΔTm  =  TmTm(x)  =  
4 σsl Tm
x ΔHf ρs
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 :
ΔTm  =  

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 K. i.e. a depression of nearly 20K in 100 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 :

x 2

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 {1}. If one normalises the distribution to unit mass of the dry porous matrix, one obtains the units {l·1·g1}. 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·g1}.

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