Theoretical Background

The distance by which x-rays penetrate into matter depends on their wavelength X and on the electron density of the atoms in the substance. In porous materials, variations in electron density between solid regions and pores cause the x-rays to scatter in a way that depends on the shape and arrangement of the pores, i.e. on the internal structure of the material. The scattering pattern from a dry porous carbon is governed both by the structure factor describing the spatial distribution of the carbon atoms in the specimen, S^q), and the electron density difference between the carbon matrix and the air-filled pores. Thus,

In Eq. (1), r0 is the classical radius of the electron (2.82 x 10-13 cm) and q = (4n/X)sin(9/2) is the magnitude of the transfer wave vector, 9 being the scattering angle. The electron density of the carbon matrix is

where Z is the atomic number, M the atomic mass, dHe the helium density of the sample and NA Avogadro's number. The electron density of the air in the pores, much smaller than pc, is neglected in Eq. (1).

If the same sample is impregnated with a liquid that does not completely fill the pore volume, then it contains three components, carbon, liquid and vapour. The scattered intensity is that of a ternary system, namely

Itern(q) = ro2{(pc-pz)2 Sd(q)+ Pc2 SCT(q)+ 2pc(pc-pl )Sv (q)} (3)

where Scl(q) is a partial structure factor describing the relative positions of carbon and liquid, Scv(q) that between carbon and vapour phase, while Scv(q) is that between liquid and vapour phase. Since the electron density of the vapour phase is much smaller than that of the liquid it is neglected. As the micropores are small and spatially uncorrelated, the effective liquid-vapour interface is in many cases also small and the third term in Eq. (3) can be overlooked. This is generally true for molecules that wet the carbon surface: at a given relative pressure they condense in the smallest pores, but merely wet the inner surface of larger pores, rather than filling them. This modifies the electron density of the liquid-filled pores, leaving the larger ones unaffected. The first two terms in Eq. (3) can then be replaced by a single pseudo-binary expression of the form

Figure 1. SAXS response from activated carbon APETA in dry air (upper curve) and in liquid hexane (lower curve). The curve for hexane vapour displays the transition from vapour phase at low q to liquid in the microporous region at q ~ 0.1 A-1 (circled).

where pS is the effective solvent density. Figure 1 shows the scattering response of an activated carbon in air, in hexane vapour and immersed in liquid hexane. At low q the intensity from the liquid hexane-containing sample is below that in air, owing to the lower contrast (pC - pS). At high q, since the hexane molecules cannot penetrate into the smallest spaces, pS = 0 and the curves converge.

The degree to which the pores appear to be filled thus depends on the spatial scale of the observation, i.e. it is q-dependent. Equation (4) is an average over the regions in the sample where the fluid is condensed and those where it is still vapour. As S1(q) and S2(q) both describe the same total structure factor of the carbon, we set S1(q) = S2(q). If the pores were uniformly filled then the two signals I1(q) and I2(q) would be everywhere proportional to each other, i.e. pS is independent of q. To account for a non uniform distribution, e.g. molecules condense only in the finest pores while remaining in the vapour state elsewhere, the effective electron density of the fluid is expressed as pS(q) = p(q)pi, where pl is the bulk density of the liquid and p(q) is the relative density of the fluid with respect to its bulk value. The filling can then be expressed by the ratio u(q) = I2(q)/Ii(q) = [pc-p(q)pl]2/pc2, (5)

In principle, 0 < p(q) <1, but p(q) may be somewhat greater than unity in the case of strong interaction between the surface and the adsorbed layer. Negative values of p(q), however, signify that the system no longer corresponds to the pseudo-binary model of Eqs. (4) and (6) and that the third term in Eq (3) is contributing to the scattering intensity. This happens when the contact area between the condensed solvent and its vapour becomes large, such as when liquid clusters develop.

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