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Thermoporometry

Thermoporometry (TPM) is a calorimetric method for the characterization of meso-porous materials with pore sizes between 2 and 50 nm [1]. When a liquid is confined in meso pores, the liquid surface is no longer planar and has a curvature. This situation causes a lowering of the triple point temperature of the liquid. This temperature depression depends inversely on the curvature radius. Therefore, a precise calorimeter such as a differential scanning calorimeter (DSC) can detect this temperature depression and the pore size distribution of the sample is obtained. One can calculate other porosity parameters such as the pore volume (Vp), internal surface area (Sp) and average pore size (rp) from the pore size distribution curve i.e. differential pore volume versus pore size [2-6].

A full thermodynamic description of TPM has been presented by Brun et al based on the Gibbs-Duhem fundamental equation [6]. In a recent study by Landy [7], the calibration equation between temperature reduction and pore radius was obtained using a well characterized controlled-pore glass (CPG). Brun’s and Landry’s methods were completely different but their results were approximately similar.

Theoretical background

Water in a meso-porous bed is divided into three parts. Free or bulk water which melts and/or freezes at 0 ºC (T0), pore water which melts and/or freezes at a few degrees below 0 ºC (T) and non-freezable water which does not participate in any phase transition. The temperature depression (ΔT=T–T0) in the pores is due to the fact that the water surface in the meso-pores is not planar but has a curvature. Brun et al [6] derived a relation between ΔT and the pore radius for the melting of ice inside the pores from the Gibbs-Duhem equation, by considering the pores as cylinders with spherical surface as Eq. (1):

rp = -32.33/ΔT + 0.68         '''Eq. (1)'''

where the thickness of non-freezable water is 0.68 nm in Eq. (1). In order to transform the melting DSC thermograph of a frozen porous sample to a pore size distribution, Eq. (2) is employed:

dVp/drp = (dQ/dt).(dt/dΔT).(dΔT/drp)/(mΔHf(T)ρ(T))         '''Eq. (2)'''

where dQ/dt is the heat flow, dt/d(ΔT) is the reverse of the scanning rate and m is the mass of dry porous sample. Also, ΔHf(T) and ρ(T) are the temperature-dependent heat of fusion and ice density for the pore water, respectively. Parameter d(ΔT)/drp is derived from Eq. (1). The functions ΔHf(T) and ρ(T) are presented in Eq. (3) and Eq. (4):

ΔHf(T) = -0.155(T-T0)2 - 11.39(T-T0) - 332         for 0>ΔT>-25 ºC          '''Eq. (3)'''

ρ(T) = 0.917(1.032 - 0.000117T)         '''Eq. (4)'''

References

[1] M.J. Van Bommel, C.W. Den Engelsen, J.C. Van Mil Tenburg, J. Porous Mat. 4 (1997) 143-150.

[2] G.W. Scherer, J. Colloid Interface Sci. 76-77 (1998) 321-339.

[3] I. Beurroies, R. Denoyel, P. Llewellyn, J. Rouquerol, Thermochim Acta 421 (2004) 11-18.

[4] T. Yamamoto, S.R. Mukai, K. Nitta, H. Tamon, A. Endo, T. Ohmori, M. Nakaiwa, Thermochim Acta 439 (2005) 74-79.

[5] M. Wulff, Thermochim Acta 419 (2004) 291-294.

[6] M. Brun, A. Lallemand, J.-F. Quinson, C. Eyraud, Thermochim Acta 21 (1997) 59-88.

[7] M.R. Landry, Thermochim Acta 433 (2005) 27-50.